C-NRLF 


*B    530    185 


SOME  GENERAT.IZATTONS  IN  THE 
THEORY  OP  SUMMABLE  DIVEEGENT  SERIES 


BT 


LLOYD  LEROY  SMAIL  , 


DISSERTATION 

Submitted  in  Pakti^ll  Fulfilment  of  the  Requirements  for  the  Degree 

OF  DocjTOR  OF  Philosophy,  in  the  Faculty  of  Pure  Science, 

Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRINTINQ  COMPANV 

LANCASTER,  PA. 

1913 


GIFT  OF 


SOME  GENEKALIZATIONS  IN  THE 
THEORY  OF  SUMMABLE  DIVERGENT  SERIES 


/ 


BT 

LLOYD  LEROY\SMAIL 


DISSERTATION 

Submitted  in  Partial  Fulfilment  op  the  Requirements  fob  the  Degree 

OF  Doctor  of  Philosophy,  in  the  Faculty  of  Pure  Science, 

Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRtNTINO  COMPANY 

LANCASTER,  Pa. 

1913 


INTRODUCTION. 

In  this  paper,  a  process  is  given  which  leads  to  four  general  methods  of 
summation  of  divergent  series,  and  each  of  these  methods  includes  as  special 
cases  several  of  the  known  methods.  These  latter  are  sufficiently  indicated 
with  their  connections  in  the  course  of  the  discussion.  It  is  shown  that, 
in  accordance  with  these  general  methods,  every  convergent  series  is  sum- 
mable  and  the  generalized  sum  is  equal  to  the  ordinary  sum;  whilst  a  properiy 
divergent  series  is  not  summable  by  these  methods.  Uniform  summability, 
and  the  continuity  of  uniformly  summable  series  and  their  term  by  term 
integration  and  differentiation,  are  discussed.  Of  the  general  theorems  ob- 
tained, applications  are  made  to  the  particular  methods  of  Cesaro,  Riesz, 
BoREL,  LeRoy,  and  the  so-called  Cesaro-Riesz  methods  of  Hardy  and  Chap- 
man. The  methods  of  proof  employed  throughout  are  simpler  than  those 
hitherto  used.  In  this  way  the  essential  properties  of  the  various  known 
methods  are  brought  out,  and  greater  uniformity  of  treatment  is  secured. 


S 


s, 


lU 


478370 


SOME  GENERALIZATIONS  IN  THE 
THEORY  OF  SUMMABLE  DIVERGENT  SERIES 


CHAPTER  I. 

The  Problem  of  Divergent  Series. 

1.  Let  us  first  seek  what  meaning  can  be  assigned  to  an  infinite  series.  In 
the  first  place  we  are  to  understand  by  an  infinite  series  a  symbol  such  as 

00 

(1)  ao  +  ai  +  02  +  •  •  •  +  a„  +  •  •  •  =  2  «n . 

To  assign  a  meaning  to  this  symbol,  the  simplest  and  most  natural  way  is  to 
form  the  expression 

n 

and  when  lim  Sn  exists,  to  take  this  value,  the  so-called  sum  of  the  series,  as 

n — ^00 

a  substitute  for  the  series  wherever  the  series  occurs  in  calculations.  Thus  the 
method  of  convergent  series  is  simply  a  particular  method  of  associating  a 
definite  number  with  the  series,  and  using  this  number  in  place  of  the  series 
in  calculations.    This  limit,  lim  *„,  however,  only  exists  for  certain  series, 

while  there  are  many  series,  the  so-called  divergent  series,  for  which  this  limit 
does  not  exist.  In  order  to  be  able  to  use  such  series,  we  must  then  find  some 
method  by  which  we  can  associate  a  definite  number  with  the  series,  so  that 
we  can  use  this  number  in  place  of  the  series  in  calculations. 

Our  problem  of  divergent  series  is  then  to  associate  with  such  a  series  a 
number,  which  we  call  the  Sum*  of  the  series,  which  should  be  defined  in  such 
a  way  that  the  resulting  laws  of  calculation  agree  as  far  as  possible  with  those 
of  convergent  series.  If  the  series  has  variable  terms,  we  wish  to  associate 
with  it,  not  a  number,  but  a  function,  which  shall  satisfy  the  above  condition. 
Any  definite  method  by  which  we  can  associate  with  a  given  series  a  Sum  is 
called  a  method  of  summation. 

2.  Chapman t  has  stated  a  "general  principle  of  summability  "  for  any 


v 


*  For  convenience,  we  write  the  sum  of  a  convergent  series,  in  the  usual  sense,  with  small 
8  ,  while  this  associated  number  here  referred  to,  we  write  Sum,  with  capital  S.  \y 

t  Quarterly  Joum.  of  Maih.,  Vol.  43,  p.  4. 

1 


/ 


-,  J  «  J/  3  ♦,..  '_     ■ 
2  LLOYD   L.   SMAIL:  SOME  GENERALIZATIONS   IN  THE 

infinite  form  as  follows:  "When  the  sequence  of  finite  forms,  which  defines 
or  generates  an  infinite  form,  does  not  tend  to  a  limit  as  the  variables  tend  to 
infinity  in  the  assigned  order  through  the  sequences  of  values  constituting  the 
domains  of  these  variables,  then  we  may  agree  that  the  number  represented 
by  the  given  infinite  form  is  to  be  the  limit  of  a  sequence  of  associated  finite 
forms,  different  from  the  members  of  the  original  sequence;  the  second  sequence 
must  of  course  be  judiciously  chosen,  so  that  the  limit  to  which  it  tends  is 
usefully  related  to  the  original  sequence.  The  number  of  its  variables  may  be 
the  same  or  greater  than  the  original  number;  and  the  additional  variables, 
if  any,  may  or  may  not  be  required  to  tend  to  infinity." 

Applying  this  to  infinite  series,  we  shall  form  sequences  involving  the  terms 
of  the  given  series,  but  which  have  limits  even  when  the  sequence  Sq,  Si, 
*2 ,  ' ' '  f  Sn,  • '  •  has  no  limit.  We  shall  not  only  form  subsidiary  sequences 
depending  upon  one  index,  but  also  sequences  depending  upon  two  indices 
(double  sequences),  and  have  simple  and  double  limits  of  such  sequences. 
An  infinite  series  will  then  be  said  to  be  summahle  by  any  particular  method  if 
the  corresponding  subsidiary  sequence  has  a  limit. 

If  the  terms  of  the  given  series  are  functions  of  a  variable  u ,  then  this  series 
will  be  said  to  be  uniformly  summahle  with  respect  to  u  in  an  interval  ( a ,  6 ) , 
if  the  subsidiary  sequence  converges  to  its  limit  uniformly  with  respect  to  u 
in  that  interval. 

3.  Methods  of  summation  have  been  classified  by  Chapman*  into  yarametric 
and  non-parametric.  A  method  of  summation  is  parametric  if  the  associated 
sequence  used  to  determine  the  Sum  contains  a  parameter  of  some  kind  upon 
the  values  of  which  depend  whether  or  not  the  series  is  summable.  If  the 
associated  sequence  contains  no  such  parameter,  the  method  is  called  non- 
parametric. 

4.  As  already  explained,  the  Sum  given  by  a  method  of  summation  is  to  be 
used  whenever  the  given  series  is  such  that  Sn  has  no  limit.  Now  «„  has  a 
limit  for  convergent  series  (in  which  case  Sn  has  a  finite  limit)  and  for  the 
properly  divergent  series  (where  «„  has  the  limit  +  »  or  —  «  ) ;  our  methods 
of  summation  are  then  designed  primarily  for  the  oscillating  series.  But  con- 
vergent series  and  oscillating  series  may  occur  in  the  same  piece  of  work,  and 
so  our  methods  of  summation  should  be  such  that  they  will  apply  to  convergent 
series  and  give  a  Sum  equal  to  the  ordinary  sum.  Likewise,  when  the  methods 
of  summation  are  applied  to  a  properly  divergent  series,  they  should  give  a 
Sum  equal  to  +  oo  or  —  <»  ,  according  as  *»  has  the  limit  +  oo  or  —  oo  . 

We  formulate  these  requirements  in  the  following  conditions,  which  we  call 
the  conditions  of  consistency: 

(1)  When  the  given  series  is  convergent,  the  Sum  given  by  any  method  of 
summation  must  exist  and  coincide  with  the  sum.     Moreover,  if  the  method 

*  Quarterly  Journ.  of  Math.,  Vol.  43,  p.  8. 


THEORY  OF  SXJMMABLE  DIVERGENT  SERIES.  3 

is  parametric,  the  Sum  must  exist  and  coincide  with  the  sum  for  all  values  of 
the  parameter  for  which  the  method  is  applicable; 

(2)  When  the  given  series  is  properly  divergent,  the  Sum  given  by  any 
method  of  summation  must  be  +  oo  or  —  oo  ,  according  as  lim  *„  is  +  «> 
or  —  00 ; 

(3)  When  the  given  series  is  uniformly  convergent,  it  must  be  uniformly 
summable;  and  this  must  be  so  for  every  value  of  the  parameter  if  the  method 
is  parametric. 


CHAPTER  II. 
Some  General  Methods  of  Summation. 
5.  Having  given  a  divergent  series  ^a„ ,  we  start  with  the  expression 

u 

[n] 

(2)  Zo./,, 

where  n  is  always  taken  positive,  and  [  n  ]  denotes  the  greatest  integer  ^  n , 
and/»  is  a  function  of  certain  variables  and  parameters  to  be  specified  presently. 
We  shall  study  the  limiting  properties  of  this  expression,  and  find  out  under 
what  circumstances  these  limits  will  satisfy  the  conditions  of  consistency. 

We  first  enunciate  some  of  the  cases  which  may  occur  and  which  we  shall 
study: 

I.  fv  may  be  a  function  of  n  and  also  of  a  parameter  k;  then,  keeping  k 
fixed,  we  seek  the  simple  limit 

[n] 

(3)  lim  ^Ovfvin,  k) . 

n-^oo   ti=0 

II.  /,  may  be  a  function  of  a  variable  x;  we  then  seek  the  repeated  double 

limits 

n 

(4)  lim  lim  2 Or /»  (a;),* 

n 

(5)  lim  lim  52  o» /t>  ( a; ) , 
or  the  Pringsheim  double  limit 

n 

(6)  lim    Sa„/v(ar), 

n,  X— >so  11=0 

where  n ,  x  tend  to  oo  independently  but  simultaneously. 

III.  /„  may  be  a  function  of  two  variables  n  and  p;  we  then  seek  the  repeated 
double  limits 

(7)  lim   lim  51  Or  /•  ( n ,  p ) , 

n    >flo  p — >oo  »=o 

(8)  lim    lim  J^a^fv  (n,  p), 

p — >«  n^->ao  ti=0 

*  The  case  x ->  oo  may  be  regarded  as  a  general  case,  since  any  other,  x—*a,  can  be  reduced 
to  this  by  transformation. 

4 


THEORY  OF  8UMMABLE  DIVERGENT  SERIES.  5 

or  we  may  seek  the  Pringsheim  double  limit     • 

fn] 

(9)  lim    52 «v/*  (w»  V)y 

n,  p— >«  t>=0 

or  we  may  seek  the  double  limit  along  a  path  F 

(10)  lim    Va„/„  (n,  p), 

where  n ,  p  tend  to  <»  simultaneously  though  not  independently,  but  in  such  a 
way  that  a  functional  relation 

F(n,p)  =  0 
holds  between  them.* 

In  place  of  starting  with  the  expression  (2),  we  may  start  with  the  expression 

(11)  jls^f., 

»=0 

« 

where  5p  =  JZ  <*»  •    Then  we  have  the  case 

i=0 

IV.  fv  may  be  a  function  of  x ,  and  we  seek  the  repeated  double  limits 

(12)  lim   lim  i:*„/.  (x), 

n — >«  X    >oo  r=0 
n 

(13)  lim   Wm  ^  Sv  fv  {x) . 

Let  us  now  examine  each  of  these  cases  more  in  detail. 

Case  I. 

6.  Let  fv  {n,  k)  he  defined  for  all  positive  integral  values  of  v ,  including  0 , 
for  all  positive  values  of  n ,  and  for  a  certain  range  of  real  values  of  the  param- 
eter k . 

Suppose  further  that/»  {n,  k)  satisfies  the  following  conditions: 

1°  0<_fv(n,k)^l        (or  every  V ,  n ,  k; 

2**  when  n  and  k  are  fixed,  the  sequence 

fOf  flf  fz)    '  '  '  }fv  ,    '  '  ' 

is  monotonic  decreasing; 
3**  lim/p  (n,  k)  =  I  for  d  fixed; 

n    >  X 

4"  foin,k)  =  l; 

5'  lim /[nj(n,  k)  =  0. 


*  See  Hardy  and  Chapman,  Quarterly  Joum.  of  Math.,  Vol.  42,  p.  187. 


6  LLOTD  L.  8MAIL:  SOME  GENERALIZATIONS  IN  THE 

When  limit  (3)  exists  and  has  the  value  S: 

(30  \\mX.a,fAn,k)  =  S, 

00 

then  we  shall  say  that  the  series  ^Ovis  summable  (/)  with  Sum  S . 

0 

We  now  proceed  to  show  that  this  method  of  summation  satisfies  the  con- 
ditions of  consistency.    It  may  be  noted  that  this  is  a  parametric  method. 

oe 

7.  Theorem  1.    If  the  series  2J  a*  w  convergent  with  sum  s ,  then 


(3) 


(»i 


lim  ^Ovfv  (n,  A:) 


n—^tn  r=0 


exists  and  is  equal  to  s  for  every  value  of  k  for  which  fv  is  defined;  so  that  every 
convergent  series  is  summable  (I)  with  Sum  equal  to  sum,  and  part  (1)  of  the 
conditions  of  consistency  is  satisfied. 
Put 

1  -fv  (n,  k)  =  gr,  (n,  k) , 

then  by  condition  1°,  we  have 

0^  gv  (n,  k)  ^  1, 
by  2°  the  sequence  go,  gi,  gz,  "  - ,  gv,  •  •  •  is  monotonic,  and  by  3°, 

lim  gv  (n,  k)  =  0 . 

n — >oo 

Put 

Gin,k)  =2lav9v{n,k)  =  2lav  -  Zlovfv  (n,k). 

v=0  0=0  r=0 

Then  we  have  to  prove  that 

(a)  limG(n,Jk)  =  0. 

« — >ao 

We  may  write 

(^)  G(n,k)  =  ^a„gr,-{-   Jl   (h  Qv 

Let  us  consider  first  the  second  sigma  of  (6).    By  Abel's  lemma,* 


[»] 


2   OvOv  {n,k) 


<  Ag  <  A, 


where  A  is  the  upper  bound  of 


23    Ov 

*  See  Bromwich,  Theory  of  Infinite  Series,  §  23. 


THEORT  OF  SUMMABLE  DIVERGENT  SERIES. 


for 

r^  N+l,  N  +  2,  .-.,  [n], 

and  g  is  the  upper  bound  of  the  Qv  for 

v=  N+1,  •••,  [n]; 

by  condition  1°,  ^  ^  1 .  Now  since  S  Op  is  convergent,  we  can  choose  No  so 
that  A  <  €  /  2  (or  N  >  Nq,  where  e  is  any  arbitrarily  small  positive  number. 
Hence 


(c) 


v=N+l 


(hgv 


foT  N>  No,n>  N  +  1. 

Now  consider  the  first  sigma  of  (6).     By  condition  3°,  when  N  is  once  fixed, 
for  every  given  positive  number  e',  we  can  find  an  integer  rriv  such  that  for 

g„{n,k)<€'         foru=  0,  1,  2,   ..-,  iV. 

Let  m  be  the  greatest  of  the  finite  set  of  integers  viq,  mi,  •  •  • ,  m^;  then  for 
n>  m,  each  gv  {n,  k)  <  e'  {v  =  0,  --  - ,  N) . 


Choose 
then 


e: 

»=0 


^Ovgv  (w,  k) 


y  If 

^12\av\  •  gv(n,k)  <  €'J2\av 


for  n  >  m . 


e'< 


i)=0 


2  El  a. 


zlcLvgv 


€ 

<2 


From  (c)  and  (d),  we  now  see  that 

\G{n,k)\  <  e 


where  N'  is  >  NqOt  m. 
Hence 


f  or  n  >  m . 


for  n  >  N', 


w 


[n] 


lim  z2  (hfv  {n,  k)  =  lim  E  «»  =  « . 

n-»ao  ti=0  n-^oo  i>=0 

The  above  argument  has  not  explicitly  involved  the  value  of  ^ ,  so  that  the 
theorem  is  true  for  every  k  for  which  /„  satisfies  the  conditions  of  §  6 . 

It  may  be  remarked  that  all  the  conditions  of  §  6  are  not  required  for  this 
proof;  we  need  only  conditions  1°,  3",  and  2°  can  be  replaced  by  the  require- 
ment that  the  sequence  (/» )  be  monotonic. 

00 

8.  Theorem  2.    If  the  series  z2  (h  (u)  is  uniformly  convergent  with  respect 


[n] 


to  u  in  the  closed  interval  (a,  b) ,  with  sum  s{u) ,  then  ^av{u)fv{n,  k)  tends 


8 


LLOYD   L.   SMAIL:   SOME   GENERALIZATIONS   IN  THE 


to  its  limit  s  (u)  uniformly  with  respect  to  u  in  the  interval  (a,  b);  so  that  c/ 
uniformly  convergent  series  is  also  uniformly  summable  (I),  and  part  (3)  of  the 
conditions  of  consistency  is  satisfied. 

Using  the  notation  of  §  7,  we  must  prove  that  G  (n,  k,  w)  -»  0  uniformly 
in  ( a ,  b) .    We  find,  as  before, 


2  <^ 


»=JV+1 


(w)  •  <7„  (n,  k) 


<  A{u)  •  g  <A  (w). 


but  since  S  a„  ( w )   is   uniformly   convergent,   we   can   find   No  such   that 
A  (u)  <  €  /  2  foT  N  >  Nq  and  for  every  m  in  (a,  6) .    We  then  have 


(c) 


22   Ov  (u)  gv  {n,  k) 


<l 


for  N  >  No  and  for  every  w  in  ( a ,  6 ) 
As  before,  we  find 


So*  (w)  •  gv  (n,  k) 


I 

e=0 


<  e'Slot,  (w) 


f  or  n  >  m , 


Let  K  be  the  upper  bound  of 


f or  w  in  ( a ,  6 ) ;  take 

then 
id) 


S  I  Ot.  ( w ) 


€< 


S  Ot,  (w)  •  ^t.  (n,  fc) 


e: 

ti=0 


<l 


for  n>  m, 


and  for  every  m  in  ( a ,  6 ) . 
Hence  G(n,A;,w)— »0  uniformly  with  respect  to  u ,  and  our  theorem  follows. 
9.  Theorem  3.    If  the  series  ^a^  is  properly  divergent,  so  that 


lim  5n  =  +  «  , 


then 


[n] 


lim  X a„/,  ( n ,  k)  =  +  oo  ; 

N-^ae  vssO 


hence  a  properly  divergent  series  is  not  summable  (I)  with  finite  Sum,  and  part 
(2)  of  the  conditions  of  consistency  is  satisfied. 


THEORY  OF  8UMMABLE  DIVERGENT  SERIES.  9 

We  have 

Z^ttvfv  =  Sofo  +  (Si  —  So)fl  +  (S2  —  Si)  fi  +    •  "   +  (*[„]   —  5[„]_i)/[„] 
v=.0 

=  5o  (/o  — /l  )  +  5l  (/l  — /2  )  + H  *[n]-l  (/[n]-l  " /[n]  )+*[n]/[n]  • 

[n]  [nl-1 

(14)  .  ■  .  S  at>/t)  =     S    *»  (/t.  —  fv+l  )   +  «[n]  /fn]  . 

»=0  f=0 

Since 

lim  5n  =  +  00  , 

if  ii  is  any  arbitrarily  large  number,  we  can  find  m  such  that  we  can  put 

Sv  =  K  -\-  ry  ioT  V  >  m,  where  r„  >  0. 

Put 

Then  (14)  becomes 

[n]  m  [n]-l 

^avfv{n,  k)  =  '^Svhv{n,  k)  -jr  K  "^   hv{n,k) 

v=0  v=0  t)=m+l 

[w]— I  m 

+    S  ryh„{n,k)+Sin]fin]in,k)  =  ^{Sv  — K)hv{n,k) 

[nJ-1  [n]_l 

H-X  S  ^t>(n,  k)  +    S  r„^„  (n,.^)  +  5[n]/rn]  (w,  A;). 
But 

Z    ^v(n,^)=   (/o-/a)+(/i-/2)+    .-.    +(/[n]-l-/r„])   =/o -/[«]. 
[i»3  m 

(a)     .'.  ^avfvin,  k)  =  J2  (Sv  —  K)  hv(n,  k)  +  K  {fo{n,  k)  —  f^n]  {n,k)] 

+    S   r„hv  {n,k) -\- Sin]f[n](n,k). 

By  condition  2°, 

hv  (n,  k)  is  ^  0; 
by  3°, 

lim  A^  (n,  k)  =  0; 

n— >oo 

byl°, 

/[n]  (n,  A:)  >  0. 

Using  these  results,  and  conditions  4",  5°,  we  get 

[n]  [n]-l 

lim  2^ Ovfy  (n,  k)  =  K  -\-  lim    J^   r«  A^  ( w ,  ^ )  +  lim  *[„] /[„]  ( n ,  A; ) . 

>    X. 

Since  K  is  arbitrarily  large,  our  theorem  follows  at  once. 


10  LLOYD    L.    SMAIL:   SOME   GENERALIZATIONS   IN  THE 

For  this  proof,  all  the  conditions  l°-5°  of  §  6  are  required. 

OB 

10.  Theorem  4.    If  the  series  ^a^  {u)  is  uniformly  summahle  (I)  with 

respect  to  u  in  an  interval  {a,  b) ,  with  Sum  S  (u) ,  and  if  the  terms  a„  ( w )  are 
continuous  functions  of  u  in  {a,  b) ,  then  S  (u)  is  a  continuous  function  of  u 
in  (o, 6) . 
Since 

[n1  00 

S  (u)  =  lim  S«»  {u)fv  (n,  k)  =  S^v  (m)/v  {n,  k) , 

n-^w  r=0  0 

it  follows  that  S  ( w )  is  the  sum  of  a  uniformly  convergent  series  of  continuous 
functions,  and  hence,  by  the  properties  of  uniformly  convergent  series,  S  {u) 
is  continuous. 

The  theorem  may  be  proved  directly  as  follows:* 
Put 

[n] 

22  a„  {u)fv  {n,  k)  =  Sn(u) . 

t>=0 

We  have 

\S{u+h)-S{u)\S\S{u-\-h)-Sn{ui-h)\-\-\Sn{u-\-h) 

-Sn(u)\'\-\Sniu)-S{u)\. 

Now  since 

lim/S„(w)  =  S{u) 

n->oo 

and  the  approach  is  uniform  with  respect  to  w  in  (o,  &) ,  we  can  find  N  such 
that  f or  n  >  iV , 


and 


\S(U)    -Sn{u)\   <|, 


Siu-^h)-SAu  +  h)\<~, 


where  e  is  any  arbitrarily  small  positive  number. 

Again,  since  Sn  {u)  is  a  sum  of  a  finite  number  of  continuous  functions,  we 
can  find  d  such  that 

\Sn{u-\-h)-Sn{u)\<^  for  U|<5. 

Hence  we  obtain 

\S{u+  h)-  S{u)\  <  €  for  I  A|  <  5, 

which  shows  that  S  {u)  is  continuous. 

ao 

11.  Theorem  5.     If  the  series  2lciv{u)  is  uniformly  summable  (I)  with 

0 

respect  to  u  in  the  closed  interval  {a,  6 ) ,  with  Sum  S  (u) ,  and  if  {ci,  c^)  is 
*  See  Chapman,  Quarterly  Journ.  of  Math.,  Vol.  43,  p.  12. 


But 


THEOKY   OF  SUMMABLE  DIVERGENT  SERIES.  11 

an  interval  contained  in  (a,b) ,  and  if  the  terms  av  (u)  are  integrahle  in  ( Ci ,  C2 ) , 
then  the  series  obtained  by  integrating  the  given  series  term  by  term  over  the  range 

S{u)  du. 
We  can  write 

where  8n  (u)  is  such  that  we  can  find  N  so  that  for  n  >  N , 

I  5„  (  W  )  I   <   €  , 

however  small  e  may  be,  for  every  uin  {a,b)  and 

[n] 

Sn  (w)  =  ^ttv  {u)fv  (n,  k). 

/»cj  /»ci  /»rj 

.'.  I     S  (u)  du  =   I     Sniu)du-\-  I     8n  (u)  du 

VCl  *J  C\  VCl 

=  ^fv{n,k)-   I     ttv  (u)  du -\-   j     8n{u)du. 

I       5n(w)rfw^     I       I  5n  (W)  I  •  I  <Zw  |<  €  (C2  —  Ci),       (n>N)y 

.'.  lim   I     5„  (w)  dw  =  0, 
lim2/p(7i,A;)   I     a,,(w)<fw=    I     S{u)du. 

n->oo  t)=0  t/cj  «/c] 

Hence  the  series  J^  I     av  {u)du\?,  summable  (I)  with  Sum   I     S{u)du* 

0      «/ci  t/ci 

Observe  that  this  proof  holds  for  any  function  f^  (n,  k) ,  such  that  Sn  (u) 
approaches  a  limit  S  {u)  uniformly,  even  though  it  should  fail  to  satisfy  all 
the  conditions  l^-b"  of  §  6. 

12.  Theorem  6.    //  the  series  ^av{u)  is  summable  (I)  with  Sum  S  (u) , 

0 

X 

then  if  the  series  zJ  <^»  ( w )  obtained  by  differentiating  term  by  term  the  given  series 

0 

is  uniformly  summable  (I)  with  respect  to  u  in  an  interval  ( a ,  6 ) ,  with  Sum 
<T  (u) ,  we  have 

a{u)  =  ^Siu). 


and  we  have 

[n] 


*  See  Chapman,  Quarterly  Journ.  of  Math.,  Vol.  43,  pp.  12-13. 


12  LLOYD   L.   SMAIL:   SOME   GENERALIZATIONS  IN  THE 

By  the  preceding  theorem,  the  series 

2  1    a„  {u)  du, 
where  c\  and  u  are  in  the  interval  ( a ,  6 ) ,  is  summable  (I)  with  Sum 

I    1^  (u)  du. 

VCl 

That  is, 

[n]  /»«  /»u 

lim  2/r  (n.  A;)   I    a,  (u)  du  =    I    cr  {u)  du; 
or  since 


JQb  {u)  du  =  ttv  (u)  —  a„  ( ci ) , 
Cl 

(a)  lim  2  {a„  (w)  —  a„  (ci)}/»  (n,  A;)  =   I    (r(w)(fM. 


we  have 


But  since  J^  a»  ( w )  is  summable  (I)  with  Sum  S  {u) ,  the  left  hand  side  of 

u 

( a )  is  equal  to  *S  (  m  )  —  S(  Ci ) ,  or 

(6)  r  a{u)du=  S{u)  -  S(ci). 

This  equation  shows  that 

Case  II. 

13.  Let  fv  {x)    be    defined    for    every    positive    value    of    x,    and    for 
v  =  0,  1,  2,   •... 

Suppose  further  that/„  {x)  satisfies  the  following  conditions: 

1°  0  ^  fv  {x)  ^  I  for  every  v,  x; 

2°  when  x  is  fixed,  the  sequence /o ,  /i ,  •••,/»,  •  •  •  is  decreasing; 

3®  '  lim/c(a;)  =  l  for « fixed; 

4°  lim  lim/„(a:)  =  0. 

14.  We  shall  first  show  that  limits  (4)  and  (6)  cannot  give  rise  to  methods 
of  summation  of  non-convergent  series. 

We  have,  by  condition  3", 

n  n 

lim  ^ttvfv  (x)  =  2 Or  > 

«— ?oo  0=0  ti=0 


THEORY  OF  SUMMABLE  DIVERGENT  SERIES.  13 

and 

n  n 

lim  lim  ^Ovfv  (x)  =  lim  ^ a» . 

OB 

Hence,  limit  (4)  can  only  exist  when  the  series  ^  a„  is  convergent,  and  will  not 

0 

give  a  method  of  summation  of  non-convergent  series.    We  need  then  consider 
limit  (4)  no  further. 

From  the  well-known  theorem:* 
"  If  lim  Up,  q  exists  and  if  lim  ttp,  q  exists  for  every  q,  then 

p,  q-^co  J>-»ao 

lim  (limOp,  g)  =   lim  ap,  q," 

q-^ao    p^oo  p,q— ^00 

we  see  that  if  lim  Op,  q  exists  for  every  q ,  then  lim  ap,  q  can  only  exist  when 
lim  lim  ap,  ,  exists. 

n 

Then  since  lim  ^  a„  /^  ( a; )  exists  for  every  n ,  it  follows  that  the  Pringsheim 
double  limit 

n 

(6)  lim   2I«»/»  (x) 

n,  x—^ao  v=0 

can  only  exist  when  the  repeated  double  limit 

n 

(4)  lim  lim  zlcivfvix) 

exists,  i.  e.,  only  when  the  series  S  av  is  convergent.     Hence  the  limit  (6) 
will  not  apply  to  non-convergent  series,  and  we  need  not  consider  it  further. 

15.  When  the  limit  (5)  exists  and  is  equal  to  S , 

n 

(5')  lim  lim  ]Ca»/»  (a;)  =  S, 

z— >aa  n— >ao  t»=0 

we  shall  say  that  the  series  2  a^  is  summahle  {II)  with  Sum  <S . 

We  shall  now  show  that  this  method  of  summation  satisfies  the  conditions 
of  consistency. 

ao 

16.  Theorem  7.    If  the  series  ^a^is  convergent  vnth  sum  s ,  then 

0 

(5)  lim  lim  X!<^r/r  (x) 

exists  and  is  equal  to  s;  that  is,  every  convergent  series  is  summahle  (H)  with  Sum 
equal  to  sum,  and  so  part  (1)  of  the  conditions  of  consistency  is  satisfied. 
*  See  Nielsen,  Lehrbuch  der  unendlichen  Reihen,  p.  76. 


14 


LLOYD  L.  SMAIL:  SOME  0ENERAUZATI0N8  IN  THE 


Put 

1  -fAx)--g.M; 

then  from  conditions  1',  2*,  3®,  it  follows  that 

for  every  v,  x,  that  the  sequence  go,  git  -"  >  gp*  '"  »s  monotonic,  and  that 

lim  g^  (x)  =  0 

for  V  fixed. 
Put 

then  since 

we  must  prove  that 


••0 

n  n 

G{n,x)  =  Y,av  -  Hovfv  (ar), 


lim  lim  G  (n,  «)  =  0. 

We  shall  first  show  that  the  Pringsheim  double  limit 

lim   G(n,x)  =  0. 


Writing  (?  ( n ,  X )  in  the  form 

(?(n,x)  =  So,^,  (x)  +  S  a»^»(a;), 
we  first  apply  Abel's  lemma  to  the  second  sigma,  and  obtain 

E  fl»y«(«)    "^  ^  '  g  "^  ^  for  every  X, 


where  A  is  the  upper  bound  of 


for    r  ==  iV  +  1 ,  •  •  • ,  n , 


and  i^  is  the  upper  bound  of  the  terms  g,  for 

c  =  iV  +  1 ,  •  •  • ,  n , 
so  that  g^\.    Since  2  a»  is  convergent,  we  can  choose  N^  so  that,  for  every 
«,  for  iV  >  iVo,  we  have  A  <  t/2;  hence 


(fl) 

for  N  >  No  and  for  every  x . 


2  a»y.  («) 


<l 


THEORY  OF  8UMMABLE  DIVERGENT  SERIES. 


15 


Now  keeping  N  fixed,  for  any  given  e' ,  by  condition  3°  we  can  find  an  in- 
teger Xv  such  that  for  a:  >  X^ ,  we  have  g,  {x)  <  t'  for  v  =  0 ,  1 ,  "  • ,  N . 
Let  X  be  the  greatest  of  the  finite  set  of  integers  Xv,  (v  =  0 ,  1 ,  •  •  • ,  N) , 
then  for  X  >  Z ,  we  have  each  g^  (x)  <  e'  {v  =  0 ,  '  ■  • ,  N) . 


Har,gvix)    ^J2\av\  '  gv(x)<  e'  -^lov 

r=0 


fora;>  X. 


If  we  take 


we  get 
(b) 


€< 


2Ek 


2a„flr„  {x) 


<\ 


From  (a)  and  (6),  we  now  obtain 

\G{n,  x)\<  t 


so  that 


f  or  a;  >  X . 


forn>  iVoH-  1,  x>  Z, 


lim  G  (n,  x)  =  0 

n,  at— >oo 


We  now  apply  the  theorem  (already  quoted  in  §  14) : 

*//   lim  ttp,  q  exists  and  lim  Op,  ,  exists  for  every  q,  then 
p,  ?->«  p-»« 

lim  lim  ap,  ,  =  lim  Up,  g." 

5— >oo  i>— >w  p,  7— >« 


Making  use  of  the  theorem:*  A  convergent  series  remains  convergent  if  its 
terms  are  each  multiplied  by  factors  which  form  a  bounded  monotonic  sequence, 

we  see  that  the  series  Z^  a^  g^^  ( x )  is  convergent  and  lim  G{n,x)  exists.     Hence 


by  the  theorem  just  quoted,  we  have 
and 


lim  lim  G  {n,  x)  =  0 , 


lim  lim  2^  «» /»  ( a; )  =  lim  2l(h>  =  s. 

Not  all  the  conditions  of  §  13  were  used  in  this  proof;  we  need  only  1",  3", 
and  in  place  of  2°  we  need  only  require  the  sequence  (/« )  to  be  monotonic. 

30 

17.  Theorem  8.    If  the  series  ^  av  is  properly  divergent,  so  that 


lim  5n  =  +  00  , 

n->QO 


•  See  Bromwich,  Theory  of  Infinite  Series,  §  19. 


16  LLOYD  L.   SMAIL:  SOME  GENERALIZATIONS  IN  THE 

then 

n 

Hm  Hm  5Zoc/p  (x)  =  +  00  ; 

that  is,  no  properly  divergent  serjj^  can  he  summable  (II)  loith  finite  Sum,  and 
hence  part  (2)  of  the  conditions  of  consistency  is  satisfied. 

Starting  with  equation  (14)  of  §9,  and  proceeding  exactly  as  in  §9,  we 
arrive  at  the  equation  corresponding  to  (a) : 

(a)         Za.fv(x)  =  Z{s.-K)Kix)  +  K{fo{x)-fnix)] 

«=0  «=0 

n-1 

+      H     Tr,  h  (X)  -{-  Snfn  {x)  , 

where 

K{x)  =fv(x)  -/h.i(«). 

We  find  from  the  conditions  of  §  13,  that 

lim  hv  (x)  =  0,         lim  fo  (x)  =  1 ,         lim  lim  fn  (x)  =  0 . 

n  n— 1 

.*.  lim  lim  ^a^fv  (x)  =  K  -\-  lim  lim  ^r„  hv  (x)  +  lim  lim  Snfn  (x) . 
But  since  r„  >  0 ,  A»  ( a; )  >  0 ,  and  *«  >  0 ,  /«  ( a: )  ^  0 ,  we  have 

n-l 

lim  lim  ^rvhg  (x)  >  0,        lim  lim  Snfn  (x)  >  0 , 

n 

.*.  lim  lim  2Zor/»  (x)  >  i^. 

z— >oo  n->oo  tt=U 

Since  K  can  be  taken  as  large  as  we  please,  our  theorem  follows  at  once. 

18.  Before  proceeding  to  show  that  this  method  of  summation  satisfies 
part  (3)  of  the  conditions  of  consistency,  it  will  be  necessary  to  make  the 
notion  of  uniform  summability  (II)  more  precise;  for  this  notion  involves 
uniform  approach  to  a  repeated  double  limit. 

A  definition  of  uniform  approach  to  a  Pringsheim  double  limit  is  easy  to 
formulate.  If  Cp,  ,  (w)  is  a  function  of  u,  and  if  for  each  value  of  u  in  an 
interval  ( a ,  b) ,  the  Pringsheim  double  limit 

lim    Op,  5  (w) 

exists  and  is  equal  to  a  ( w ) ,  then  we  shall  say  that  flp,  «  ( w ) ,  approaches  its 
Pringsheim  double  limit  a  (u)  uniformly  with  respect  to  u  in  the  interval  ( a,  6 ), 
if  for  any  positive  «,  two  integers  P,  Q  can  be  found  such  that  for  every 


p^P,q^Q, 


THEORY   OF  SUMMABLE  DIVERGENT  SERIES. 


\a  (U)  —  ttp,  q{u)\   <   € 


17 


for  every  value  of  w  in  ( a ,  6 ) . 

19.  Theorem  9.    If  cip,  q  (u)  approaches  its  Pringsheim  double  limit 

lim    Up,  q  (u)  =  a  (u) 

uniformly  with  respect  to  u  in  an  interval  (a,  6) ,  and  if  for  each  q,  ap,  q  (u) 
approaches  a  simple  limit 

lim  ap,  q(u)  =  ag  (u) 

uniformly  with  respect  to  u  in  {a,  b) ,  then  aq{u)  approaches  the  simple  limit 

lim  aq{u)  =  a{u) 

uniformly  with  respect  to  u  in  (a ,  b) . 

By  hypothesis,  we  can  find  numbers  P ,  Q  such  that 


(a) 


I  a  (u)  —  ap,  q  (u)  I  < 


toT  p>  P ,  q  >  Q,  and  for  every  w  in  ( a ,  6 ) ;  also  we  can  find  a  number  Pq 
(depending  in  general  upon  q )  such  that 


(6) 


lap.  5  (w)  -  ttq  (u)  I  <2 


for  p>  P'q  and  for  every  u  in  {a,  b) .    Adding  (a)  and  (6),  we  get 

I  a  (w)    —   ttq  (u)  I   <   € 

foT  q>  Q  and  for  every  u  in  ( a ,  b) ,  which  gives  the  result  stated  in  the 
theorem. 

20.  The  theorem  just  proved  suggests  a  definition  of  uniform  approach  to  a 
repeated  double  limit,  which  will  be  useful  in  our  later  discussion  of  uniform 
summability. 

If  for  each  value  of  u  in  the  interval  ( a ,  b) ,  the  repeated  double  limit 

lim  lim  ap,  q  (u) 

exists  and  is  equal  to  a  (u) ,  then  we  shall  say  that  flp.  «  (w)  approaches  its 
repeated  double  limit  a  (u)  uniformly  with  respect  to  w  in  ( a ,  6 ),  if  ap,  q  (u) 
approaches  its  simple  limit 

lim  ap,  q(u)  =  aq  (u) 


1§  LLOYD   L.    SMAIL:   SOME   GENERALIZATIONS  IN  THE 

uniformly  with  respect  to  w  in  (a,  b) ,  and  if  a,  (w)  approaches  its  limit 

lim  ttg  (u)  =  a  (u) 
3^00 

uniformly  with  respect  to  w  in  ( a ,  6 ) . 

n 

Our  definition  of  uniform  summability  (II)  will  then  be:  If  2Za»  (w)/„  (a;) 

t;=0 

approaches  its  repeated  double  limit 

n 

lim  lim  ^a„{u)fv{x)  =  S  {u) 

I— ^00  n— >so  v=0 

OB 

uniformly  with  respect  to  u  in  an  interval  ( a ,  6 ) ,  then  the  series  5Z «»  ( w ) 

0 

is  said  to  be  uniformly  summable  (II)  with  respect  to  w  in  {a,  b) . 

21.  We  are  now  in  a  position  to  discuss  the  uniform  summability  (II)  of  a 
uniformly  convergent  series;  but  before  entering  upon  that,  we  shall  first 
prove  an  auxiliary  theorem,  in  order  not  to  interrupt  the  argument  later. 

Theorem  10.    A  uniformly  convergent  series^ av  (u)  remains  uniformly 

0 

convergent  if  its  terms  are  each  multiplied  by  factors  gr„ ,  provided  that  the  sequence 

(gv)  is  monoionic,  and  that  \  gv\  <,  a  constant  c* 

Since  the  sequence  {gv)  is  monotonic  and  \gv\  <  c ,  g,,  must  approach  a 

limit,  call  it  g .     Put  bv  =  g  —  gv  when  ( gr„ )  is  an  increasing  sequence,  and 

K  =  gv  —  g  when  (gv)  is  decreasing.    Then  the  sequence  (bv)  is  monotonic 

decreasing  and  approaches  the  limit  0 .     Since 

Ov  (w)  •  gv  =  av  (u)  '  g  —  av  (u)  •  6„    or    av  (w)  -  g  +  a„  (u)  •  bv, 

oe 

we  need  only  prove  thatZ]a»  (u)  -  bv  is  uniformly  convergent.    If  A  {u) 

0 

is  the  upper  bound  of 

r 

for 

r  =  m+  1>  •••,  m  +  p, 

we  have  by  Abel's  lemma 

m+p 

'^aviu)-bv   <  A{u)  ■  bm^i<  Aiu)  ■  bo. 

But  since  JIcLv  (u)  is  uniformly  convergent,  we  can  find  M  such  that  for 
0 

m>  M ,  we  have 


A{u)<j- 


*  This  b  a  generalization  of  the  theorem  of  §  19  in  Bromwich,  Theory  of  Infinite  Series. 


THEORY  OF  SUMMABLE  DIVERGENT  SERIES.  19 

however  small  e  may  be,  for  every  u  in  an  interval  ( a ,  6 ) . 


m+p 

^•ttv  (u)  •  bv 

TO+1 


<   € 


toT  m>  M  and  for  every  positive  integer  p ,  and  for  every  u  in  (a,  b) . 

00 

Hence  the  series  ^Ov  (u)  b^is  uniformly  convergent  in  (a,  b) . 

0 

oe 

22.  Theorem  11.     If  the  series  ^aviu)  is  uniformly  convergent  with  respect 

0 

n 

to  u  in  the  interval  (a,  b) ,  with  sum  s  (u) ,  then  23a„(w)  •/,,  (x)  tends  to  its 

repeated  double  limit  s{u)  uniformly  with  respect  to  u  in  (a,  b);  that  is,  a  uni- 
formly convergent  series  is  also  uniformly  summable  (II),  and  so  part  (3)  of  the 
conditions  of  consistency  is  satisfied. 

n 

We  shall  first  show  that  X)«d(w)  '  fv(x)  approaches  its  Pringsheim  double 

v-O 

limit  s  (u)  uniformly  with  respect  to  w  in  ( a ,  6 ) . 

Using  the  notation  of  §  16,  we  have  to  prove  that  numbers  N ,  X  can  be 
found  such  that  for  every  n>  N ,  x  >  X , 

I  G  (n,  X ,  u)  \  <  € 
for  every  u  in  {a,  b) . 
As  in  §  16,  we  find  by  Abel's  lemma 


2   av  (u)  gv  (x) 


^  A(u)  •  g  S  A{u), 
for  every  x  and  for  every  u ,  where  A  (u)  is  the  upper  bound  of 


X)a»(w)|     for    r=iV+l,---,n, 

and  g  is  the  upper  bound  of  the  terms  gv  ior  v  =  N  -\-  1 ,   •  •  • ,  n .     Since 
zl^v  (u)  is  uniformly  convergent,  we  can  find  iVo  such  that  for  N  >  No,  we 

0 

have  ^(w)<e/2for  every  w  in  ( a ,  6 ) . 

(a)  .'.I   S   dv  (u)  gv  (a;)|<  ^ 

for  N  >  No,  for  every  x ,  and  for  every  w  in  ( a ,  6 ) . 
As  before,  (§  16), 

\^av  (u)  gv  {x)\<  e'Yl\civ(u)\       for  every  x>  X . 

I  »=0  I  «=0 


20  LLOYD  L.  smail:  some  generalizations  in  the 

Let  K  be  the  upper  bound  of  S  |  a„  ( w )  |  for  w  in  ( o ,  6 ) ;  and  take  e'  <  ^. 

Then 

(6)  Jlav(u)gv{x) 

»=0 


€ 

<2 


for  a;  >  Z  and  for  every  w  in  (a,  b) . 
Hence 

n  I 

J^civ  {u)g^{x)\<  e 

»=o  I 

(oT  n>  N  >  No,  x>  X ,  and  for  every  u  in  ( a ,  b);  that  is,  G  (n,  x,  u) 
approaches  its  Pringsheim  double  limit  0  uniformly. 

Qvix)  evidently  satisfies  the  conditions  of  Theorem  10,  so  that,  since  Sa»(w) 

is  uniformly  convergent,  ^Uv  (u)  -  g^,  (x)  is  also  uniformly  convergent;  that 

0 

is,  the  limit 

n 

lim  Ylciv  (u)  gv  {x) 

is  approached  uniformly  with  respect  to  w  in  ( a ,  6 ) .     Now  applying  Theorem 
9,  we  see  that  the  limit 

n 

lim  lim  Z^a„(w)^„(a;) 

X — >ao  n— ^00  v=0 

is  approached  uniformly  with  respect  to  u  in  ( a ,  6 ) .     Our  theorem  then 
follows. 

23.  Theorem  12.     //  the  series  2J«»  (w)  is  uniformly  summable  (II)  with 

0 

respect  to  u  in  an  interval  ( a ,  6 ) ,  with  Sum  S  (u) ,  and  if  the  terms  a„  ( w )  are 
continuous  functions  of  u  in  {a ,  b) ,  then  S  (u)  is  a  continuous  function  of  u 
in  {a,b) . 
Put 

and 

then 


Z)a«  (w)/«  (x)  =  Sn.  X  (w), 


limSn.  X  (w)  =  Sx(u), 

n->ge 

lim5x(w)  =  S{u). 


We  have 


\S{u+h)-Siu)\^\Siu-\-h)-SAu  +  h)\  +  \SAu  +  h) 

(a)  -   Sn.  X  (W  +  A)  I  +   I  Sn,  X  (W  +  A)  -  Sn.  x  (w)  | 

+   I  -Sn.  X  (W)  -  -Sx  (W)  I  +  !  Sx  (W)  -S{U)\. 


(d)  \Sn,Au)-SAu)\  <^, 


THEORY   OF  SUMMABLE   DIVERGENT  SERIES.  21 

From  the  definition  of  uniform  summability  (II),  it  follows  that,  for  any 
given  € ,  we  can  find  X  such  that  for  x  >  Z ,  we  have 

(b)  \SAu)-Siu)\  <^, 
and 

(c)  \S{u  +  h)-SAu+h)\<^^, 
and  that  we  can  find  N  such  that  f or  n  >  N , 

(d) 
and 

(e)  \SAu-hh)- Sn,x{u-[-h)\<^. 

Since  Sn,  x  (u)  is  a.  sum  of  a  finite  number  of  continuous  functions,  we  can 
find  5  such  that  for  |  A  |  <  5 ,  we  have 

Combining  (6)-(/) ,  we  get 

\S{u-{-  h)  -  S{u)\<  e  for|A|<5, 

from  which  the  theorem  follows. 
24.  Theorem  13.    If  ^Ov  (u)  is  uniformly  summable  (II)  with  respect  to 

0 

u  in  an  interval  ( a ,  6 ) ,  with  Sum  S  (u) ,  and  if  its  terms  a„  (u)  are  integrable, 
then  the  series  obtained  by  integrating  the  given  series  term  by  term  with  respect  to 

S(u)du. 

Using  the  notation  of  §  23,  since 

lim5„.  x{u)  =  Sx{u) 

n->oo 

and  the  approach  is  uniform  with  respect  to  w  in  ( a ,  6 ) ,  and  since 

00 

limS„,  X  (w)  =  Hov  (w)/»  (x), 

n— >ao  0 

it  follows  that  the  series 

Sx{u)  =  J^a„  (u)f„  (x) 

0 

is  uniformly  convergent  in  ( o ,  6 ) .     Hence 

I     Sx{u)  du  =  ^fr,  (x)  I     aviu)du. 

*Jex  0  *J  ci 


22  LLOYD  L.   8MAIL:  SOME  GENERALIZATIONS  IN  THE 

/.  lim  Hm  Jlfv{x)  I    a^{u)du=  lim  S/e(a;)  I     at,(w)(iw=  lim  I     Sx{u)du. 
Then,  to  prove  our  theorem,  we  must  show  that 

lim   I     (Sx(w)dw=   I     S{u)du. 


But 


We  can  write 

where  i?*  ( w )  approaches  its  limit  0 ,  as  a;  — >  oo  ,  uniformly  with  respect  to 
ti  in  ( a ,  6 ) ;  we  can  then  find  X  such  that  for  ar  >  X , 

I  '/o!  ( w")  I  <  €  for  every  m  in  ( a ,  6 ) . 

/»<•»  /*"»  /»«t 

I     S  (u)  du  =   j     Sx{u)du+  j     rfx  (u)  du, 

I       T/x  (  W  )  fZw  ^     I        I  r/x  (  W  )  I  •  I  <iw  I  <   €  (  C2  —  Ci)        f  OF   X  >   Z  . 

7ix{u)  du  =  0        and        lim    I     Sx{u)  du  =    I     S  (u)  du. 

«0 

25.  Theorem  14.    If  23  a„  (w)  i*  summable  (II)  m<A  Swm  5  (w),  am?  if 

0 
w 

<Ae  5ene5  ^al  (u)  obtained  by  differentiating  the  given  series  term  by  term  with 

0 

respect  to  u  is  uniformly  summable  (II),  with  Sum  a  {u) ,  then 

a{u)=^S{u). 
The  proof  is  practically  the  same  as  that  of  §  12. 

Case  III. 

26.  Let  /„  ( n ,  p )  be  defined  for  all  positive  values  of  n ,  p ,  and  for 

«  =  0,  1,  •••,  [n], 
and  let 

/„  ( n ,  p )  =  0  for  v>  n. 

Suppose  that/»  (n,  p)  satisfies  the  following  conditions: 

1°  0^/e(n,p)^l  for  every  D,  n,  p; 

2°  when  n ,  p  are  fixed,  the  sequence  /o ,  /i ,  ••-,/«,  •  •  •  is  decreasing; 
3"  lim/„  (n,  p)  =  1 ,  and  when  N  has  been  fixed,  we  can  choose  no  so  that 


THEORY  OF  SUMMABLE   DIVERGENT  SERIES.  23 

fv  {n,  p)  -*l  uniformly  for 

V  =  0 ,  1 ,  -  ■  • ,  N ,  n  "k  no] 
4*  lim  lim  /„  ( n ,  p )  =  1  f or  «  fixed ; 

p— ^00  n— >» 

5°-  lim   fv  (n,  p)  =  1    for  «  fixed,  for  certain  paths  F; 

F 

6®  lim  lim  /[«]  (n,p)  =  0; 

7°  lim   /[„]  {n,  p)  =  0  for  certain  paths  F. 

n,p— >« 
F 

Hardy  and  Chapman*  have  shown  that  the  limits  (7)  and  (9)  can  only 
exist  when  the  given  series  2  a„  is  convergent,  so  that  they  will  not  give  rise 
to  methods  of  summation  of  non-convergent  series.  We  need  not  consider 
them  further. 

When  the  limit  (8)  exists  and  is  equal  to  S : 

[n] 

(8')  lim  lim  Yj civ fv  (n,  p)  =  S , 

p—^con-^aov=0 

we  shall  say  that  the  series  2  a^  is  summable  (III  A)  with  Sum  S . 
When  the  limit  (10)  exists,  with  the  value  S: 

(10')  lim    ^ttvfv  (n,  p)  =  S , 

n,  p— >oo  v=0 
F 

we  shall  say  that  the  series  S  a„  is  summable  (III  B)  with  Sum  S . 

We  shall  now  show  that  both  these  methods  satisfy  the  conditions  of  con- 
sistency. 

X 

27.  Theorem  15.    If  the  series  ^a^  is  convergent  with  sum  s ,  then  it  is  sum- 

0 

mable  (III  A)  and  also  summable  (III  B)  with  Sum  equal  to  s  in  both  cases,  so 
that  part  (I)  of  the  conditions  of  consistency  is  satisfied  for  both  methods. 

Hardy  and  Chapman  have  proved  f  that  when  2  a„  is  convergent  with  sum 
s ,  then 

lim    zlovfv  (n,  p)  =  s. 
It  follows  at  once  that  the  Hmit  (10)  taken  along  any  path  F  will  be  equal  to  s . 

00 

Since  /»  ( n ,  p )  ^  1 ,   the  series   ^  Ovfv  (n,  p)   is  convergent,   so  that 

0 

X/  Ovfv  ( n , 
in  §  14 ,  we  see  that  the  limit  (8)  exists  and  is  equal  to  s 


lim  z2 dvfv  (n,  p)  exists.     Then  from  the  theorem  on  double  limits  quoted 

n— ^00   r=0 


*  Quarterly  Joum.  of  Math.,  Vol.  42  (1911),  p.  202. 
t  Quarterly  Joum.  of  Math.,  Vol.  42  (1911),  p.  202. 


24 


LLOYD  L.   SMAIL:   SOME  GENERALIZATIONS  IN  THE 


28.  Theorem  16.    If  the  series  X  a^  (m)  is  uniformly  convergent,  with  sum 

0 

8  {u) ,  in  an  interval  (a,  6) ,  then  it  is  also  uniformly  summable  (III  A)  and 
(III  B)  with  respect  to  u  in  {a,  b) ,  and  part  (3)  of  the  conditions  of  consistency 
is  satisfied  for  both  methods. 

[n] 

We  shall  first  prove  that  ^a„(u)fv{n,p)  approaches  its  Pringsheim  double 

c=0 

limit  s  (u)  uniformly  with  respect  to  w  in  ( a ,  6 ) . 
If  we  put 

1  —fv  {n,  p)  =  Qv  {n,  p), 
and 

[nl 

G(n,p,u)  =  ^av(u)  g„{n,p), 


we  must  prove  that  numbers  N ,  P  can  be  found  such  that  for  n>  N  and 
p>  P ,  we  have 

\G  {n,  p,u)\  <  € 
for  every  w  in  ( a ,  6 ) . 
Just  as  in  §  22,  we  have 


2-j   Ov 


v=y+i 


{u)gv  {n,p) 


<:  A  (u)  '  g  <  A  (m), 


for  every  p  and  u ,  where  A  (u)  and  g  are  defined  as  before.    Since  X!  «r  ( w ) 

u 

is  uniformly  convergent,  we  can  find  A^o  such  that  A  (u)  <  e / 2,  and  there- 
fore 


(a) 


N+l 


{u)g„  {n,p) 


<l 


for  N  >  No,  for  every  p ,  and  for  every  w  in  ( a ,  6 ) . 

Now  when  N  is  fixed,  by  condition  3°  we  can  choose  no,  po  such  that 
\  gv  {n ,  p)  \  <  e'  ioT  V  =  0,  1,  "  • ,  N ,  n  >  no ,  p  >  Pol  hence 


Jl  a„  (u)  g,,  {n ,  p)   <€'2]|a,(w)|     (n>no,p>po). 


Let  K  be  the  upper  bound  of  J^  |  a„  ( w )  |  for  w  in  ( a ,  6 ) ,  and  take  e'  <  e  /  2K . 
Then 


(6) 


A 

E 

t=0 


Z)ar  (u)  gv  {n,  p) 


<2 


for  n  >  no,  p>  po,  and  for  every  w  in  ( a ,  6 ) . 
(a)  and  (b)  then  give 

(c)  \G{n,p,u)\  <  € 

f  or  n  >  N' ,  p>  Po,  and  for  every  w  in  ( o ,  6 ) ,  where  N'  is  the  greater  of  No 
and  no- 


THEORY  OF  SUMMABLE  DIVERGENT  SERIES.  25 

It  follows  at  once  from  the  result  just  proved  that  £  a,  ( w ) /„  ( n ,  p)  ap- 
proaches  its  double  limit  along  a  path  F: 

[n] 

lim    ^ av  (u)  f„  (n ,  p)  =  s  (u) 

F 

uniformly  with  respect  to  m  in  ( a ,  6 ) ;  so  that  the  part  of  our  theorem  which 
relates  to  the  method  (III  B )  is  proved. 

Using  Theorem   10,  we  can  readily  show,  as  in  §  22,  that  as  n  — >  oo  , 
M 
zL  (iv  {u) fv  {n ,  p)  approaches  a  limit  uniformly  with  respect  to  w  in  (a,  h) . 

t.=0 

Then  by  Theorem  9,  it  follows  from  the  result  obtained  in  the  first  part  of 
this  section,  that  X^a„(w)/«(n,p)  approaches  its  repeated  double  limit 

11=0 

[n] 

lim  lim  Xa„(w)/„(n,p)  =  s  {u) 

p— >ao  n — ^00   r=0 

uniformly  with  respect  to  w  in  ( a ,  6 ) ;  so  that  the  part  of  our  theorem  which 
refers  to  the  method  (III  A)  is  proved. 

00 

29.  Theorem  17.     If  the  series  zlctv  is  properly  divergent,  with  lim  5„=  +  oo , 

0  n->oo 

then  it  is  not  summable  (III  A)  or  (III  B)  with  finite  Sum;  hence  part  (2)  of 
the  conditions  of  consistency  is  satisfied. 
Equation  (a)  of  §  9  becomes  in  this  case: 

[n]  m 

(a)     J2a„f^{n,  p)  =  Y<  (Sy  -  K)  h  {n ,  p)  -\-  K  {fo  {n ,  p)  —  f^]  {n ,  p)  } 

•=0  »=0 

+    S  r,>hv{n,p)  +  sin]f[n]{n,p), 

where  h  (n,  p)  =  fy  {n,  p)  —  /„+ 1  ( n ,  p) . 
From  conditions  4°,  6°,  we  have 

lim  lim  ^  (n,  p)  =  0,     lim  lim/o  (n,  p)  =  1,    lim  lim/[n]  (n,  p)  =  0. 

p->oon— >«  p— >aon->oo  |}— >oo  n — >« 

Hence 

W  [n]-l 

(6)     lim  lim  2_/a»/»  (w,  p)  =  jfiT  +  lim  lim    ^   ryh„{n,p) 

p— >«  n— >w  r=0  p— ^00  n— >oo  v=m+l 

+  lim  lim5[n]/in]  in,p). 

p-^oo  n— ^00 

But  since  r„  >  0,  K  (n,  p)  >  0  (by  2"),  5[„]  >  0,  /[n]  (n,  p)  >  0,  the  last 


2Q  LLOYD  L.  smail:  some  generalizations  in  the 

two  terms  on  the  right  in  (6)  are  positive,  so  that 


lira  lim  Z^avfv(n,p)>K. 

p— ^00  n— >w  «=0 

Hence  S  a,  is  not  summable  (III  A)  with  finite  Sum. 
Again,  from  conditions  5°,  7",  we  have 

lim  Ap  (n,  p)  =  0,    \im  fo(n,p)  =  1,    lim /[„]  (n,  p)  =  0. 
Also 


n,  D-^ao  n,  p— >oo  n,  o— >oo 

F  F  F 


lim      zl  rr,K{n,'p)>0,      lim  «[„]/[«]  (n,  p)  >  0; 
z'  r 

hence 


2^  a 


lim   22^r,fv  (n,  p)  >  /ii. 


n,p-^oo  „—Q 
F 


and  therefore  S  a»  is  not  summable  (III  B)  with  finite  Sum. 
30.  Theorem  18.    If  ^  av  (u)  is  uniformly  summable  (III  A)  or  (III  B) 

0 

with  respect  to  u  in  an  interval  {a,  b)  with  Sum  S  (u) ,  and  if  the  terms  o^  ( m ) 
are  continuous  functions  of  u  in  (a,  b) ,  then  S  (u)  is  continuous  in  ( a ,  b) . 

The  proof  of  the  part  of  the  theorem  relating  to  the  method  (III  A)  is 
precisely  similar  to  that  of  Theorem  12. 

To  prove  the  other  part,  put- 

Zlav{u)fv{u,p)  =  Sn,p{u), 
then  Sn,  p  (u)  approaches  its  limit 


lim  Sn,  p  (u)  =  S  (u) 

n,p-^m 
F 

uniformly  with  respect  to  w  in  ( a ,  6 ) ,  so  that 

(O)  |S(«)-Sn,p(w)|<^ 

for  every  w  in  ( a ,  6 ) ,  if  n ,  p  are  so  chosen  (satisfying  the  relation  F(n,  p)  =  0) 
that  the  corresponding  point  ( n ,  p )  is  sufl5ciently  far  along  the  path  F . 
Writing 

\Siu+h)-  S(u)\^\S{u+h)-Sn,piu+h)\ 

+  \Sn.p{u+h)-   Sn.p{u)\-\-\Sn,p{u)-S{u)\, 

the  first  and  last  terms  on  the  right  are,  by  (a),  each  <  €  /  3  for  proper  choice 


THEORY  OF  8UMMABLE   DIVERGENT   SERIES.  27 

of  n,  p,  and  the  second  term  can  be  made  <  «  /  3  by  taking  \  h\  <  5;  hence 

\S{u+h)  -  S(u)\<  e  for  I  A|  <  5, 

and  our  theorem  follows. 

00 

31.  Theorem  19.     If  the  series  zLciviu)  is  uniformly  summable  (III  A)  or 

0 

(III  B)  with  respect  to  u  in  (a,  b) ,  with  Sum  S  (u) ,  and  if  the  terms  a»  (m) 
are  integrable,  then  the  series  obtained  by  integrating  the  given  series  term  by  term 
toith  respect  to  u  over  a  range  (cy,  Ca )  included  in  ( o ,  b)  is  summable  (III  A) 

or  (III  B)  respectively,  with  Sum   I     S  (u)  du. 

The  proof  of  the  part  referring  to  (III  A)  is  the  same  as  that  of  Theorem  13. 
To  prove  the  other  part,  let  us  write 

S  {u)  =  Sn,  p  (u)  +  5n.  p  (m), 

where  8n,  p  (u)  approaches  its  limit 


lim  8n.  p  (u)  =  0 

«,p->ao 

uniformly  with  respect  to  w  in  ( a ,  6 ) . 

S{u)du=    I     Sn,p{u)du-\-   I     dn,  p{u)  du, 
and 

r[n]  ^ 

Sn,  p  (w)  du  =  2/r  (n,  p)  I     a»  (w)  du, 
•=0  Jei 

By  properly  choosing  n,  p,  we  can  make 

I  5n,  p  (W)  I   <   € 

for  every  w  in  ( a ,  6 ) ; 

K,  p{u)  du\<^    \     \hn,  p{u)\\du\<  e{c2  —  Cx) 

for  n,  p  properly  chosen. 

.'.    lim     I     8n,  p  (u)du  =  0, 

n,p— >oo  t/cj 
F 

and 

[n]  /»c,  pct 

lim   Ylfv(.n,p)  I     av  (u)  du  =    I     S  (u)  du. 

n,p--^«  v=0  Jci  Jci 

00 

32.  Theorem  20.    IfY^a^iu)  is  summable  (III  A)  or  (III  B)  with  Sum 

0 

00 

S  (u) ,  and  if  the  series  z2  a^  ( w )  obtained  by  term  by  term  differentiation  of  the 


28  LLOYD  L.  smail:  some  generalizations  in  the 

given  series  is  uniformly  summable  (III  A)  or  (III  B)  respectively,  with  Sum 
<r  ( w ) ,  then 

aiu)=-^Siu). 

The  proof  is  the  same  as  that  of  §  12 . 

Case  IV. 

33.  Let  fv  (x)  be  defined  for  all  positive  values  of  x  and  for  all  positive 
integral  values  of  v ,  including  0 . 

Suppose  also  that  /» ( a; )  satisfies  the  following  conditions : 

1°  fv  (x)  >  0  for  every  v,  x; 

2°  lim/„(a:)  =  0  for  t>  fixed; 

OS 

3**  ^fv  (x)  is  convergent  for  every  x,  and 

0 

00 

\\m  J2  fv  (x)  =  1 . 

»  »->oe  t»=0 

The  limit  (12)  cannot  be  used  for  the  summation  of  non-convergent  series 
(nor  even  for  convergent  series),  for  by  2°, 

n 

lim  Ylsvfv  (x)  =  0, 

z— >ao  t»=0 

and  therefore 

(120  lim  lim  12sJ,{x)  =  0. 

When  the  limit  (13)  exists  and  is  equal  to  S , 

(130  lim  lim  J2s,fAx)  =  S, 

ae— >ao  n—^oo  v=0 

we  shall  say  that  the  series  S  a«  is  summable  (IV)  with  Sum  S . 

00 

34.  Theorem  21.    If  the  series  ^a„  is  convergent,  vrith  sum  s,  then 

0 

n 

lim  lim  2«t»/t»  (x)  =  *; 

that  is,  every  convergent  series  is  summable  (IV)  with  Sum  equal  to  sum,  so  thta 
part  (1)  of  the  conditions  of  consistency  is  satisfied. 
Put 

Sn  =  s  •{•  8n,        where        lim  5»  =  0 . 


THEORY   OF  8UMMABLE  DIVERGENT  SERIES. 


29 


Then 
By  3°, 
so  that 
Since 


llsvfv  (x)  =  s  •  J^  fv  (x)  4-  Xl5„/„  (x). 


lim  lim  ^  fv{x)  =  1 , 

x—^ao  n— >«  v=0 
n  n 

lim  lim  ^Svfv{x)  =  s  -\-  lim  lim  ^8vfv  (x) . 

«— >w  n— >oo  »=0  z— >«  n— ^00  v=0 

lim  6„  =  0 , 


we  can  find  m  such  that  |  5»  |  <  e  for  c  >  m. 
Now 

n  n 

H^vfvix)    ^  Si  5j  •/„  (a;), 


and 


•=0 


r=)»»+l 


Now  since 
for  V  finite, 


Z    \8,\-fAx)<e   Z   /.  (a:)<  €i:/,(a:), 

v=iin-l  t»=:m+l  0 

«  Bt  «» 

.'.  Z  I  5.  I/.  (a:)<  Z  I  5.  l/t,  (a;)  +  6  Z /.  (a?) . 
lim/^Cx)  =  0 


I— >00 


lim  Z/t.  (a;)  =  1, 

X— >oo       0 

and  since  «  can  be  taken  as  small  as  we  please,  we  have 


We  have  then 


lim  lim  Z 5» /»  (x)  =  0 . 

w 

lim  lim  Z  *»  /»  ( a; )  =  * . 


35.  Theorem  22.    If  22  civ  is  properly  divergent,  so  that 

0 

lim  5n  =  +   00  , 

lim  lim  Z^c/t.  (a:)  =  +  « ; 
/Acn  part  (2)  o/  <^  conditions  of  consistency  is  satisfied. 


30 


LLOYD   L.   SMAIL:   SOME   GENERALIZATIONS  IN  THE 


As  before,  we  can  put  Sv  =  K  -\-  r„  for  v  >  m,  Tv  >  0,  where  K  is  an  ar- 
bitrarily large  number.     Then 


Since 


«=m+l 


\imfr,(x)  =  0,    Hm  lim  Z/»(x)  =  1,     r„  >  0,    fv{x)>0, 

z— ^00  s->ao  n— >co     0 

we  have 

n  n 

lim  lim  ^s^fvix)  =  X  +  lim  lim    ^   r^fv  {x)  >  K. 

X— >x  n— >0B  11=0  i-^QO  n— >ao  ii=»»+l 

00 

36.  Theorem  23.    //  the  series  ^av{u)  is  uniformly  convergent  in  a  closed 

0 

n 

interval  (a,  b) ,  with  sum  s  (u) ,  then  2J*»  (u)  fv  (x)  approaches  its  repeated 
double  limit 

n 

lim  lim  23 *«  (w)/»  (x)  =  s  (u) 

X— >oo  n— >co  ti=0 

uniformly  with  respect  to  u  in  {a,  b);  that  is,  a  uniformly  convergent  series  is 
aJ-so  uniformly  summable  (IV),  and  part  (3)  of  the  conditions  of  consistency  is 
satisfied. 
Write 

Sn  {U)   =   S  (U)  -\-  dn  (U)  , 

then  we  can  find  m  such  that  for  n  >  m , 


(a) 


8»(w)|  <  € 


for  every  w  in  ( a ,  6 ) . 

(6)  lis,  {u)f,  {x)  =  s  iu)  ilfv  (x)  +  1:5,  {u)fr,  (x). 

00 

Now  by  3°,  zlfv  (x)  is  convergent  for  all  values  of  a; ,  so  that  if  we  denote  its 

0 

sum  by  F  ( a: ) ,  we  can  find  N  such  that  ior  n>  N  and  for  x  fixed, 


Fix)-Zfv{x) 


<  €. 


Then  for  any  fixed  w  in  ( a ,  6 ) ,  we  have 

F(x)'siu)-s{u)j:fvix) 


<  €|*(w)|         (n>  N) 


THEORY   OF  8UMMABLE  DIVERGENT  SERIES.  31 

Now  since  s  (u)  is  the  sum  of  a  series  uniformly  convergent  in  the  closed 
interval  (a,  b) ,  \s  {u)\  must  have  a  finite  upper  bound,  say  K,  in  (a,  6); 
hence 


F(x)  '  s(u)-s(u)T>fAx) 


<K 


for  n>  N  and  for  every  w  in  ( a ,  6 ) .    Therefore  s  (u)  ^  f^  (x)  approaches 

0 

its  limit  s  (u)  •  F  (x)  uniformly  with  respect  to  w  in  ( a ,  6 )  as  n  — >  oo  . 
Again,  since  by  3° ,  lim  F  (x)  =  1 ,  we  can  find  X  such  that  for  a;  >  X , 

\F{x)  -  1|  <  €. 

For  any  fixed  w  in  ( a ,  6 ) ,  we  then  have 

\F  (x)  '  s(u)  -  s(u)\<  e\s{u)\  for  a;  >  X; 

therefore 

\F{x)siu)-s{u)\  <K€ 

ioT  x>  X  and  for  every  u  in  (a,  b) .    Hence  F  (x)  s  (u)  approaches  its 
limit  s  (u)  uniformly  with  respect  to  w  in  {a,  b)  as  x-^  oo  . 

n     - 

Referring  to  our  definition  (§  20),  we  see  that  s  (u)  ^fv  (x)  approaches 

0 

its  repeated  double  limit 

n 

lim  lim  s  (u)^  f^  (x)  =  s  (u) 

«— >ao  n— >so  0 

uniformly  with  respect  to  w  in  {a,  b) . 

n 

We  have  next  to  show  that 21 5,  ( m ) /„  (x)  approaches  its  repeated  double 

0 

limit 

n 

lim  lim  ^dviu)f„ix)  =  0 

uniformly  with  respect  to  w  in  ( a ,  6 ) . 
We  have 


T,8Au)fAx)-T.8Au)fvM 


By  (a). 


^E|5.(w)|/.(x) 


Sl+l 


E  I  5.  (m)  I/,  ix)  <  €Z/»  (xX  €  Zf.  (x)  <  e  Z/.  (x) 

■»+l  w+l  0  0 


for  every  urn  {a,b) .    But  X/t>  (ic)  is  convergent  for  every  x,  so  that 

0 

Fix)^tfvix) 


32  LLOYD  L.  smail:  some  generalizations  in  the 

is  bounded  for  all  values  of  x,  and  if  G  be  the  upper  bound  of  F  (z) ,  we  have 

i:\8Au)\fv{x)<G€ 

for  every  w  in  ( a ,  6 ) . 


m+l 


ll5Au)fAx)-ll8Au)fv{x) 


<G€ 


for  every  w  in  ( o ,  6 ) .    Hence  ^8v  {u)fv  (x)  approaches  its  limit,  as  n  -♦  oo  , 

0 

uniformly  with  respect  to  w  in  ( a ,  6 ) . 
We  must  now  show  that  S  5p  ( w )  /»  ( a; )  approaches  its  limit  0 ,  as  a:  ->  «  , 

0 

uniformly  with  respect  to  w  in  ( a ,  6 ) .    m  is  now  fixed.    Since 

lim/„  {x)  =  0, 


for  every  v,  we  can  find  Xv  such  that  for  a;  >  X„,  we  have/r  {x)  <  e;  let  X 
be  the  greatest  of  the  finite  set  Zo ,  Xi ,  •  •  • ,  Xm ,  then  we  have  each/^  ( a: )  <  € 
fora;>  X  {v  =  0,1,  •••,m). 

Z8.(w)/.(ar)    ^ll\K{u)\fAx)<€ll\K{u)\     {otx>X. 

0  0  0 

m 

Now  K{u)  is  bounded  in  (a,  6),  so  that  Z^|  5»  (w)  |  has  a  finite  upper 

0 

bound,  call  it  H .    Then 

0 

for  a;  >  X  and  for  every  w  in  ( a ,  6 ) .    Hence  X)5„  (w)/t,  (a;)  approaches  its 

0 

limit  0 ,  as  ar  — >  00  ,  uniformly  with  respect  to  w  in  ( a ,  6 ) . 

n 

We  have  now  shown  that  ^5»  {u)fv  (x)  approaches  its  repeated  double 

0 

limit  0  uniformly;  and  our  theorem  is  now  proved. 

00 

37.  Theorem  24.    If  ^Ov  {u)  is  uniformly  summable  (IV)  loith  respect 

e 
to  u  in  an  interval  (a,  6) ,  with  Sum  S  (u) ,  and  if  its  terms  a,  (u)  are  con- 
tinuous in  ( a ,  b)  then  S  (u)  is  continuous  in  ( a ,  b) . 

The  proof  of  this  theorem  can  be  carried  through  in  the  same  way  as  that 
of  §  23,  the  only  difference  being  that  here  <S„,  »  (w)  is  expressed  in  terms  of  *, 
instead  of  Or ,  but  since  *„  ( w )  is  a  sum  of  a  finite  number  of  continuous  functions, 
it  follows  that  Sn,  «  ( w )  is  here  a  continuous  function  of  u . 


THEORY  OF  SUMMABLE  DIVERGENT   SERIES.  33 

38.  Theorem  25.     //  ^Oviu)  is  uniformly  summahle  (IV)  with  respect 

0 

to  u  in  {a ,  b) ,  vrith  Sum  S  (u) ,  and  if  the  terms  Or,  (u)  are  integrable,  then  the 
series  obtained  by  integrating  term  by  term  the  given  series  over  a  range  ( Ci ,  C2 ) 

S  (u)  du. 
Putting 

n 

Sn,x  (U)  =   Jlsv  {U)fv  {X),  lim  Sn,  x  (u)  =   S^  (w), 

0  n— >« 

we  have 

n  00 

Sx  (u)  =  lim  ^s„  {u)f„{x)  =  Y,Sv  {u)fv  (x) , 

n— >ao     0  0 

X 

SO  that  the  series  ^Sv  {u)fv  (x)  is  uniformly  convergent,  and  we  can  in- 

0 

tegrate  term  by  term,  and  get 

^fv{x)  I     Sr,{u)du=   I     Sx{u)du, 
the  series  on  the  left  being  convergent.     Hence 

lim  lim  Z^fv  {x)  \     Sv{u)du  =  lim   I     Sx{u)du. 

Since 

limiS^(w)  =  S{u) 

and  this  approach  is  uniform  with  respect  to  w ,  by  the  method  used  in  §  24 
we  can  prove  that 

Sx  {u)  du  =    \     S  {u)  du . 

-A  •'Cl 

(a)  .*.  lim  lim  2Z/t>  (a?)   I     5„(w)(iw=    I     S{u)du. 

«— >ao  n— >oo  t)=0  »/ci  t/cj 

To  prove  that  the  series  2^  I     a„  ( w )  c?m   is   summable   (IV)   with   Sum 
I     iS(w)  dw,  we  must  show  that 

lim  lim  SiSI     Ov  (u)  du\  f^  (x)  =   j     S  (u)  du. 

a:— >oo  n-^aa  v=Q  ^  p=0  tJci  '  *Jc\ 

But 

zl  \     av  {u)  du  =   \     •{^a„(w)f<iw=l     Sv  (u)  du, 

and  hence  by  (a),  our  theorem  follows  at  once. 


34  LLOTD  L.  smail:  some  generalizations  in  the 

00 

39.  Theorem  26.    If^Or,  {u)  is  summable  (IV)  iviih  Sum  S  (u) ,  and  if 

0 

the  series  ^a,  (u)  obtained  by  term  by  term  differentiation  of  the  given  series  is 

0 

uniformly  summable  (IV)  vnth  Sum  <r  (u) ,  then 

a(u)=-^J{u). 

The  proof  is  precisely  the  same  as  that  of  Theorem  14. 

40.  Reviewing  the  general  results  obtained,  we  see  that  the  conditions  of 
consistency  are  satisfied  by  all  four  of  our  general  methods  of  summation 
(I)-(IV);  and  that  uniformly  summable  series  possess  properties  similar  to 
those  of  uniformly  convergent  series. 


CHAPTER  III. 

Particular  Methods. 

A  number  of  particular  known  methods  of  summation  are  included  as  special 
cases  under  our  methods  (I) -(IV),  and  we  shall  now  apply  our  general  theorems 
to  these. 

Case  I. 

Cesaro's  Method. 

41.  Let  the  function /„  (n,  k)  be  defined  for  all  positive  values  of  n,  for 
V  =  I,  2,  '  • ' ,  [n],  and  for  every  real  k  except  negative  integral  values,  by 
the  equation 

n(n-  I)  (n- 2)  '■•  {n- v-hl) 


(15)  fAn,k)  = 


(A;  +  n)(fc  +  n-  l)(A:  +  n-2)  ...  (k  +  n-v+l) 


and  for  v=  0,  let  /o  (n,  k)  =  1,  for  v>  n,  let  /„  (n,  k)  =  0.  We  shall 
call  this  the  Cesaro  function,  and  denote  it  hy  Cfv  {n,  k) . 

We  must  first  show  that  Cfv{n,k)  satisfies  the  conditions  of  §  6. 

It  is  easily  seen  that  3°  and  4"  are  true  for  every  k;  also  that  1°  is  true  if 
k>  0,  but  if  A;  <  0  and  n>  \k\,  then 

CfAn,k)Sl. 
Since 

Cfv+i  {n,  k)         n  —  V 


Cfv  {n,k)       k-\-  n  —  v' 

this  ratio  is  <  1  if  ^*  >  0 ,  and  is  >  1  if  A;  <  0  and  n  is  sufficiently  large,  so 
that  the  sequence  {Cfv)  is  decreasing  for  A:  >  0  and  increasing  for  A;  <  0 
and  n  large  enough;  2°  is  then  true  for  A;  >  0 .  We  shall  not  consider  values 
of  A:  <  0 . 

It  remains  only  to  show  that  5°  is  satisfied.  For  Cesaro's  method  it  is 
sufficient  to  give  n  only  positive  integral  values.  We  need  then  only  prove 
that 

lim  Cfn(n,k)  =  0. 

n—^aa 

We  have* 

n  !  n*' /I   I 

(Ar+n)(ifc+n-  1)  •••  (A;+ 1)  ~  ^"  ^^"^  ^  ^ ' 


*  See  Nielsen,  Lehrbuch  der  unendlichen  Reihen,  p.  248. 

35 


36  LLOYD  L.  smail:  some  generalizations  in  the 

where 

limrn(A;+l)  =  r(A;+l). 


.'.\{mCfn(n,k)  =  0. 

When  a  series  S  c^,  is  summable  (I)  with  the  Cesaro  function  Cfv  (n,  k) , 
it  is  usual  to  say  that  the  series  is  summable  (C ,  k) . 
The  corresponding  summation  formula  is 

rifi^       o  _  ,.      Y>         n  {n  -  1)  -  •  ■  {n  -  v-\-  1) ^ 

42.  The  Theorems  1-3  give  us  the  following  results: 

Theorem  27.  Every  convergent  series  is  summable  {C ,  k)  for  every  k  >  0, 
ivith  a  Sum  equal  to  its  sum. 

This  theorem  has  been  proved  by  CHAPMAN.f 

Theorem  28.  A  properly  divergent  series  is  not  summable  ( C ,  k)  with 
finite  Sum  for  any  value  of  k  >  0 . 

This  theorem  is  new;  it  has  been  proved  when  A:  is  a  positive  integer,  however, 
by  Nielsen,  t 

Theorem  29.  A  uniformly  convergent  series  is  uniformly  summable  {C ,  k) 
for  k>  0. 

This  has  been  proved  by  Chapman.  § 

Riesz's  Method. 

43.  Let  the  function /„  (n,  k)  be  defined  for, every  positive  value  of  n,  for 
every  real  k ,  and  for  ij  =  1,2,  •  •  • ,  [  n  ] ,  by  the  equation 

(17)  y.(„,i)={i_Mil}'. 

where  X  (n)  is  a  positive  monotonic  function  of  n,  increasing  to  «  with  n; 
f or  tj  =  0 ,  let 

fo{n,k)  =  1, 

for  this  we  must  assume  that  X  (0)  =  0;  and  for  r  >  n,  let 

fv{n,k)  =  0. 

We  shall  call  this  the  Riesz  function,  and  denote  it  by  Rfv{n,  k) . 

*  When  A;  is  a  positive  integer,  this  is  the  definition  given  by  CesXro  (BvUelin  des  sciences 
math.,  ser.  2,  Vol.  14  (1890),  pp.  114-120).  When  A;  is  any  real  number  >  —  1  ,  the  method 
has  been  discussed  by  Chapman  {Proc.  London  Math.  Soc.,  ser.  2,  Vol.  9  (1911),  pp.  369-409), 
and  by  Knopp  {Sitzungsberichte  der  Berliner  Math.  GeseU.,  Vol.  7  (1907),  pp.  1-12). 

t  Proc.  Lond.  Math.  Soc.,  ser.  2,  Vol.  9  (1911),  p.  377. 

t  Nielsen,  Elemente  der  Funktionentheorie  (1911),  pp.  194-5. 

§  Quarterly  Joum.  of  Math.,  Vol.  43,  pp.  24-25. 


THEORY   OF  SUMMABLE  DIVERGENT  SERIES.  37 

It  follows  at  once  from  the  definition  that  conditions  l*'-4''  of  §  6  are  satis- 
fied when  ^*  >  0 . 
We  shall  suppose  further  that  X  ( n )  satisfies  the  condition : 

(18)  lim  — r-7 — r—  =  1 , 


n— >« 


X(n) 


then  it  follows  that 


and 


y  X([n])  ^ 
hm  >  ,  .  = 1 
i^«  X  (n) 


n_>«  L  X  (n)    J 

so  that  condition  5°  of  §  6  is  satisfied  for  A;  >  0 . 

When  a  series  is  summable  (I)  with  the  Riesz  function  Rfv  (n,  k) ,  Hardy 
and  Chapman  call  it  summable  {R,  X,  k).*  The  corresponding  summation 
formula  is 

(19)  S=limZa.    l-fhri     • 

n-»oor=0  «■  A(n)J 

44.  The  Theorems  1-3  applied  to  this  method  give: 

Theorem  30.  A  convergent  series  is  summable  {R,  X ,  k)  for  every  k>  0, 
tcith  Sum  equal  to  sum,  if\  satisfies  the  condition  (18). 

This  is  proved  by  Chapman  and  Riesz. 

Theorem  31.  A  properly  divergent  series  cannot  be  summable  (R,  X,  k) , 
for  any  k>  0,  with  finite  Sum,  if\  satisfies  (18). 

This  result  is  new. 

Theorem  32.  A  uniformly  convergent  series  is  uniformly  summable 
(R,\,  k)  for  every  k>  0,if\  satisfies  (18). 

This  theorem  is  also  new. 

Case  II. 
LeRoy's  Method. 

45.  Let  the  function  fv  (t)  be  defined  for  all  positive  integral  values  of  v , 
including  0 ,  and  for  all  values  of  t  such  that  0  <  t  <  1 ,  by  the  equation 

T(vt+1) 


(20)  /,(0  = 


r(^+i)- 


We  shall  consider  it  for  values  of  t  near  1.    We  call  it  the  LeRoy  function 

and  denote  it  by  L/„  (<)  • 
This  function  is  positive  for  all  values  oi  v,t  under  consideration,  and  since 
♦  See  Chapman,  Proc.  Lond.  Math.  Soc.,  ser.  2,  Vol.  9  (1911),  p.  373. 


38  LLOYD  L.  smail:  some  generalizations  in  the 

T  (x)  increases  as  x  increases,  for  x  >  1.462  •  •  •  *,  we  have 

Also 

liinZ/„(0  =  1, 

and 

Lfo{i)  =  l. 

That  the  sequence  Lfo,  Lfi,  •  •  • ,  Lfp,  •  •  •  is  decreasing  may  be  shown 
as  follows: 

r(vi+t+l)      T(vt+1) 


(a)  LUUt)-LfAt)  = 


r(2)  +  2)       r(2)+i) 

T{rt+t-\-  1)  _T{vt-\-  1) 


Now 

Tivt'\-l  +  t)=   f  e-'x'^+'dx,        T{vt+l)=   Ce-'x^'dx. 

By  integration  by  parts,  we  get 

or 

I     C-' «•*+'  (ia;  =  <(»  +  1 )  I     e-'  a:'"+*-i  dx, 
Jo  Jo 

.'.  T  (vt  +  t+  I)  =  t  {v+  l)T  {vt  +  t)<  {v-\-  1)T  {vt  +  t) ,        since  <<1. 

But  r  (r/  +  0  <  r  (r<  +  1 ) ,  hence 

ib)  T{vt+t-}-l)-  {v+l)T{vt+l)<0. 

From  (a)  and  (6),  we  see  that  Lfv  decreases  as  v  increases. 
We  shall  next  show  that 

hm   ;  7;  ;  =  o. 

By  Stirling's  formula,!  we  have 


r(n+lj 
,^„e-''7i''+*v/2,r 


lim  .v:;r;-^=i, 


...           r(n<+  1)          e-*n'»+*/27r       , 
. .  lim ^^=  •  =  1 , 

»-»ooe-'»'(nO"'^l/27r      r(n+l) 


*  See  the  graph  of  r  ( z )  in  Klein,  Hypergeometrische  Funktion,  p.  122,  or  Godefeot, 
Th6orie  616mentaire  des  series,  p.  250. 

t  See  Bromwich,  Theory  of  Infinite  Series,  p.  462. 


THEORT  OF  SUMMABLE  DIVERGENT  SERIES.  39 

or 

T  {nt+  1)         e-"  n"  •  n*      _ 

^"^  iji  r  (n  +  1 )  '  e-«'  n"'  n»  r'  v'<  ~     ' 

Now 

*        '^     **  _   g-n(l-0  yjn(l-Of-««  =   g-n(l-0+n(l-0  log  n-nt  tog  t 


=   g(l-t)n  log  n-n [(l-0+<  k«  t] 

But  since  i  <  1, 

(5)  lim  g(l-On  log  n-nll-t+t  log  <]   =    _|.    qq  ^ 

n— 3>as 

From  (a)  and  (6),  we  see  that  we  must  have 

hm  ^pTT TTV  =  0 . 

«_>«  r  (w+1) 

Then 

lim  limi/rt  (0  =  0. 

If  we  now  make  the  change  of  variable  t  =  e~^'' ,  we  get  a  function  of  x: 


LfAx)  = 


T(v-\-l)   ' 


which  satisfies  all  the  conditions  of  §  13. 

When  a  series  is  summable  (II)  with  the  LeRoy  function  Lfv  (x)  ,we  shall 
say  that  the  series  is  summable  (L) .  The  corresponding  summation  formula 
is 

(21)  g  =  limlim±%7;'+^>. 

This  definition  is  one  used  by  LeRoy.* 

46.  When  we  apply  the  general  Theorems  7,  8  to  this  method,  we  have  the 
theorems: 

Theorem  33.    A  convergent  series  is  summable  (L)  mth  Sum  equal  to  sum. 
This  was  proved  by  Hardy,  t 

Theorem  34.    A  properly  divergent  series  is  not  summable  (L)  with  finite 
Sum. 
This  result  is  new. 

BoreVs  Method. 

47.  Let  the  function  fv  (x)  be  defined  for  all  positive  values  of  x ,  and  for 
every  positive  integral  value  of  v ,  including  0 ,  by  the  equation 


(22) 


BfAx)  =  f'e-f^da, 


♦  Annalea  de  Toulouse,  ser.  2,  Vol.  2  (1900),  pp.  323-327. 
t  Quarterly  Joum.  of  Math.,  Vol.  35  (1903),  pp.  36-37. 


40  LLOYD  L.  smail:  some  generalizations  in  the 

This  function  is  always  positive;  and  since  the  integrand  is  positive,  we  have 

rg~"  a"  da  <    I     e~*  a"  da  =  D  ! 

80  that  Bfv  (x)  <  1  for  every  v,  x.    Thus  condition  1°  of  §  13  is  satisfied. 
By  integration  by  parts,  we  find 

/c—  a*+^  da=  -  e-^  a"^^  +  {v -{-  l)f  e'' a"  da, 

•*•  /     ■    IN  ,        e—  a*^^  rfa  =  -  ,     ,    ,,,  +  -i        e-^a^'da, 
(»+  1)  !Jo  {v+l)l      vlJo 

or 

5/h.i  (a:)  -  5/.  (x)  =-  e--— ^ 


(1^+  1)1' 

The  right  hand  member  of  this  equation  is  always  negative,  so  that  as  v 
increases,  Bfv  (x)  decreases,  and  2**  is  satisfied. 
We  have  at  once 

lim5/„(a;)  =  1, 

and  3°  is  satisfied. 

That  4°  is  satisfied,  may  be  shown  as  follows: 
We  first  seek  the  limit 


Stirling's  formula  gives  us 


Now 

a"      e-"  n"-^  1^2^  a* 


lim  — , . 


hm ^.    . —  =  1 . 

g-n  „«+j  y/2ir 


n  1  n  1  e'"  n""'"*  /2ir  * 

so  that 

a"        .  a" 

(a)  lim  — ,  =  lim ^.    , —  . 

We  have 

a"" 


g—n  ff^n+i 


—.  gn  ^n  ij— »»— 4  =:  gw+w  •<>«  «— (n+1)  log  «  ^  g"'^"'"  •**  •!— (»»+4)  tof  « 


But 

lJ|jl'gn{l+IOga)-(n+J)IOgn  _   Q 

(6)  .Mim^=0. 


THEORY  OF  8UMMABLE  DIVERGENT  SERIES.  41 

We  can  then  find  N  such  that  a"  /  n  I  <  e  for  n  >  N . 

.•.jrc-^da<  €j     e-^da=  €{l  -  e"')         f  orn  >  N. 

.'.  lim   I    e~^—.da  =  0, 

m— >ao  »/0  W  I 

and 

lim  limfi/,  (x)  =  0, 

SO  that  4°  is  satisfied. 
All  the  conditions  of  §  13  are  therefore  satisfied  by  Borel's  function. 

The  summation  formula  for  this  method  is 

S  =  lim  lim  Yl(h  •   \    e~^—:da 

=  lim  lim    1    e~^i^av—.)da. 
If  the  series  22(h>  oc^  I  vl  defines  an  integral  function,  or  if  it  defines  an  analytic 

0 

function  which  is  susceptible  of  analytic  continuation  along  the  real  axis,  and 
therefore  has  no  singular  points  on  the  real  axis,  we  get 

S  =  lim    I    e~'^[^av—.)da. 

z-^co  Jo  \     0  '^  •  / 

(23)  .,s  =  J".-.(|:a.f,)<fe. 

This  is  Borel's  famous  integral  definition.*  When  a  series  is  summable  (II) 
with  Borel's  function/,  (x) ,  that  is,  when  the  integral  (23)  is  convergent, 
we  say  that  the  series  is  summable  {Bz). 

48.  We  find,  by  applying  Theorems  7  and  8,  the  results: 

Theorem  35.     A  convergent  series  is  summable  (B2)  with  Sum  equal  to  sum. 
This  was  proved  by  Hardy.! 

Theorem  36.     A  properly  divergent  series  is  not  summable  ( fij )  vrith  finite 
Sum. 
This  is  proved  in  Bromwich,  Theory  of  Infinite  Series,  p.  270. 

Elder's  Power  Series  Method. 

49.  The  function 

(24)  fAx)  =  e~' 

evidently  satisfies  the  conditions  l°-4*'  of  §  13. 

*  See  BoREL,  Lemons  sur  les  series  divergentes,  p.  99. 
t  Cambridge  Philoa.  Transactions,  Vol.  19,  pp.  298-299. 


42  LLOYD  L.  smail:  some  generalizations  in  the 

Our  summation  formula  here  is 

n  r 

S  =  lim  lim  zlcir,  e  ' . 

If  we  put  e  '  =  2 ,  it  becomes 

(25)  S  =  limZ;a,2«. 

«-^l      0 

This  is  sometimes  called  Euler's  power  series  method. 

Theorem  7  applied  to  this  case  gives  Abel's  theorem  on  the  continuity  of 
power  series. 

Case  III. 

The  Cesaro-Riesz  methods*  appear  here  as  a  special  case. 

50.  Let  the  function  fv  {n,  p)  be  defined  for  all  positive  values  of  n,p, 
and  for 

c  =  0,l,2,  •••,[n], 
by  the  equation 


(26)  ^.(„„)  =  {i_X-MW 


(n) 


where  X  (n)  is  a  positive  monotonic  function  increasing  to  <»  with  n,  and 
satisfying  the  conditions  X  ( 0 )  =  0  and 

..     X(n-l)      ^ 
hm      ^  .    V —  =  1 . 
n->ao     X(n) 

It  is  easily  seen  that  conditions  1°,  2°,  and  the  first  part  of  3°  of  §  26  are 
satisfied.  Hardy  and  Chapman  t  have  shown  that  the  second  part  of  3° 
is  satisfied. 


Since 


we  have 


'^  .(.) 


so  that  4"  is  satisfied. 
Since 


hm  1  1  —  ,   .    .  f       = 
r^^  L         X  (n)  J 


lim  lim  /t,  ( n ,  p )  =  1 , 

p->oo  n—^co 


X(n-l) 
hm      ^  .    . —  =  1 , 
n->oo     X(n) 


and  X(n)— >  «  asn-»  oo,we  have 

X(") 


so  that  6°  is  true. 


«->«  l  X(n)   J 


*  Called  thus  by  Hardy  and  Chapman,  Quarterly  Joum.  of  Maih.,  Vol.  42,  p.  191. 
t  Quarterly  Joum.  of  Math.,  Vol.  42,  p.  204. 


THEORT  OF  8UMMABLE  DIVERGENT  SERIES.  43 

Now  suppose  that  the  path  F  is  defined  by  the  relation 

=  w  ( n ) ,  where  lim  w(  n )  is  finite; 

then 

"•VI         X(n)J  „^«  I        X(n)J 

*'V  I  X(n)    J  «^,  I  X(n)    J 

Thus  5°  and  7°  are  also  satisfied. 

When  a  series  is  summable  (III  A)  with  the  Cesaro-Riesz  function,  we  shall 
call  it  summable  ( CR ,  X ) ,  and  when  it  is  summable  (III  B)  and  the  path 
F  is  given  by  the  function  w  ( n ) ,  we  shall  call  it  summable  (  Cil ,  X ,  « ) . 

51.  From  Theorems  15  and  17,  we  have: 

Theorem  37.  A  convergent  series  is  summahle  ( CR ,  X )  *  toiih  Sum  equal 
to  sum. 

This  theorem  is  new. 

Theorem  38.  A  convergent  series  is  summable  {CR,  X,  co)  for  every  path 
F  {as  described  in  §  50),  mith  Sum  equal  to  sum. 

This  was  proved  by  Hardy  and  Chapman.! 

Theorem  39.  A  properly  divergent  series  is  not  aumvable  {CR,  X )  nor 
{CR,\,o}). 

This  result  is  new. 

Case  IV. 

62.  Let  us  take  first 

(27)  fAx)  =  e-^ 'f-^. 

We  have  at  once  fv  {x)  >  0 , 

lim  /„  ( a: )  =  0 , 

Zfv{x)  =  e-'f:^^=e-''e'=V, 

0  0      «'  ' 

so  that  conditions  V,  2°,  3°,  of  §  33  are  all  satisfied.    The  corresponding  sum- 
mation formula  is 

(28)  S  =  lime-'E*.-;, 

which  is  Borel's  exponential  definition.  J 

When  a  series  is  summable  (IV)  with  Borel's  function  (27),  we  shall  say 
that  it  is  summable  {  Bi) . 

*  Where  X  satiafiea  the  conditions  of  §  50. 

t  Quarterly  Joum.  of  Math.,  Vol.  42,  p.  204. 

X  See  BoBEL,  Legons  sur  lea  series  divergentes,  p.  97. 


44  LLOYD   L.   8MAIL:  SOME   GENERALIZATIONS  IN  THE 

More  generally,  take 


(29)  /.(a:)  =  6-*.^*, 

where  A:  is  a  positive  integer;  this  can  be  shown  to  satisfy  the  conditions  of 
§  33  just  as  for  (27).  When  a  series  is  summable  (IV)  with  Borel's  function 
(29),  we  shall  call  it  summable  {Bz,  k) .  When  k  =  1 ,  we  have  summability 
{B\) .    The  summation  formula  is 


(30)  S=  lime-*  £5,^. 

This  is  Borel's  generalization  of  his  exponential  method.* 
If  we  take 

(31)  /.(a:)  =  e-'.^, 

it  is  easily  shown  that  this  function  satisfies  the  conditions  of  §  33.    The 
resulting  method  is  one  studied  by  CosTABEL.f 

Still  more  generally,  let  <t>  {x)  be  an  integral  function  2Z  CnX^ ,  where 

0 

Cn  >  0 .    Take 

(32)  /«(^)  =  ^  •^'^^- 

This  function  satisfies  the  conditions  of  §33,  and  includes  all  the  preceding 
functions  of  this  §. 

53.  Applying  Theorems  21,  22  to  Borel's  methods,  we  have: 
Theorem  40.     A  convergent  series  is  summable  {B\)  with  Sum  equal  to  sum. 
This  is  proved  in  Vivanti-Gutzmer,  Theorie  der  eindeutigen  analytischen 

Funktionen,  pp.  328-9. 

Theorem  41.  A  properly  divergent  series  is  not  summable  (Bi)  toith  finite  Sum. 

This  also  is  given  in  Vivanti-Gutzmer,  p.  329. 

Theorem  42.  A  convergent  series  is  summable  {Bz,  k)  with  Sum  equal  to 
sum  for  every  k . 

This  result  is  proved  in  Bromwich,  Theory  of  Infinite  Series,  pp.  300-301. 

Theorem  43.  A  properly  divergent  series  is  not  summable  {Bz,  k)  toith 
finite  Sum,  for  any  k . 

This  is  new. 

54.  The  general  theorems  of  Chapter  II  on  the  uniform  summability  of 
uniformly  convergent  series,  and  the  continuity,  integration,  and  differentia- 
tion by  uniformly  summable  series  apply  of  course  to  the  particular  methods  of 
this  chapter,  but  as  all  the  results  so  obtained  are  new,  they  have  not  been 
stated  explicitly. 

*  See  BoREL,  Legons  sur  les  series  divergentes,  p.  129. 

t  L'Enseignement  mathimatique,  Vol.  10  (1908),  pp.  387-388. 


REFERENCES. 

BoREL,  E.     Le9ons  sur  les  series  divergentes.     Paris,  1901. 

Bromwich,  T.  J.  Ta.,  Introduction  to  the  Theory  of  Infinite  Series. 
London,  1908.     (Chap.  XL) 

Cesaro,  E.,  "  Sur  la  noiultiph cation  des  series,"  Bull,  des  sc.  Math.,  ser.  2, 
Vol.  14,  pp.  114-120.     (1890.) 

Chapman,  S.,  "On  Non-Integral  Orders  of  Summability  of  Series  and  In- 
tegrals," Proc.  London  Math.  Soc,  ser.  2,  Vol.  9,  pp.  369-409.     (1911.) 

Chapman,  "On  the  General  Theory  of  Summability,  with  Applications  to 
Fourier's  and  other  Series,"  Quarterly  Journ.  of  Math.,  Vol.  43,  pp.  1-52. 
(1911.) 

CosTABEL,  A.,  "  Sur  le  prolongement  analytique  d'une  fonction  m^romorphe," 
VEnseignement  Math.,  Vol.  10,  pp.  377-390.     (1908). 

Hardy,  G.  H.,  "On  Differentiation  and  Integration  of  Divergent  Series," 
Tram.  Cambridge  Philos.  Soc,  Vol.  19,  pp.  297-321.     (1903.) 

Hardy,  "  Researches  in  the  Theory  of  Divergent  Series  and  Divergent  In- 
tegrals," Quarterly  Journ.  of  Math.,  Vol.  35,  pp.  22-66.     (1903.) 

Hardy  and  Chapman,  "  A  General  View  of  the  Theory  of  Summable  Series," 
Quarterly  Journ.  of  Math.,  Vol.  42,  pp.  181-219.     (1911.) 

Knopp,  K.,  "  Multiplication  divergenter  Reihen,"  Sitzungsherichie  der  Berliner 
Math.  Ges.,  Vol.  7,  pp.  1-12.     (1907.) 

Le  Roy,  E.,  "  Sur  les  series  divergentes  et  les  fonctions  definies  par  un  d^ 
veloppement  de  Taylor,"  Annales  de  Toulouse,  ser.  2,  Vol.  2,  pp.  317-430. 
(1900.) 

Nielsen,  N.,  Lehrbuch  der  unendlichen  Reihen.    Leipzig,  1908. 

Nielsen,  Elemente  der  Funktionentheorie.     Leipzig,  1911. 

RiESZ,  M.,  "  Sur  les  series  de  Dirichlet,"  Comytes  Rendus,  148,  pp.  1658- 
1660.     (1909.) 

RiESZ,  "  Sur  la  sommation  des  series  de  Dirichlet,"  Comptes  Rendus,  149, 
pp.  18-21.     (1909.) 

RiESZ,  "  Sur  les  series  de  Dirichlet  et  les  series  enti^res,"  Compies  Rendus, 
149,  pp.  909-912.     (1909.) 

Vivanti-Gutzmer,  Theorie  der  eindeutigen  analytischen  Funktionen.  Leip- 
zig, 1906. 


45 


VITA. 

Lloyd  Leroy  Smail.  Born,  Columbus,  Kansas,  September  23,  1888; 
entered  University  of  Washington,  1907;  Tutor  in  Mathematics,  1909-11; 
Undergraduate  Assistant  in  Mathematics,  1910-11;  A.B.,  1911;  Fellow  in 
Mathematics,  1911-12;  A.M.,  1912;  Fellow  in  Mathematics;  Columbia 
University,  1912-13;  member  of  Sigma  Xi;  member  of  the  American  Mathe- 
matical Society. 

The  writer  gratefully  acknowledges  his  indebtedness  to  Professor  W.  B. 
Fite  for  his  valuable  assistance  and  encouragement. 


46 


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